Calculation of Coefficients of the Optimal Quadrature Formulas in ${\mathit{W}}_2^{(7,0)}$ Space

Abstract

In this paper, we construct an optimal quadrature formula in the sense of Sard by Sobolev’s method in the $W_2^{(7,0)}$ space. We give explicit expressions for the corresponding optimal coefficients. This formula is exact for exponentional–trigonometric functions.

1. Introduction

It is known that in many practical situations, especially in engineering, economics, and medicine, if you want to solve some problems, you will usually need to take the integral of a function. But we have difficulty in calculating the exact values of some definite integrals. So we need to construct optimal quadrature formulas to estimate their numerical values (see [1,2,3,4,5,6]). Additionally, quadrature formulas also play an important role in calculating the numerical solution to differential and integral equations. And the accuracy of the quadrature formulas is an important problem, too. Specific applications include improving the accuracy and stability of numerical solutions when solving partial differential equations numerically. In signal processing and image processing, optimal quadrature formulas can better handle the local and global properties of signals and images, enabling functions such as noise reduction and feature extraction. In the Monte Carlo method, its theory can also be combined to optimize sampling strategies and integral estimates, improving computational efficiency and accuracy for calculating high-dimensional integrals, probability distributions, and other problems. We need to continuously improve the algorithms to make the errors smaller and smaller. The specific algorithms in this paper can also be applied. For example, to construct an optimal quadrature formula, it is necessary to determine the optimal coefficients by minimizing the norm of the error functional. In numerical integration algorithms, the error estimation theory in Sobolev spaces can be used to improve the algorithm and enhance its accuracy. In function interpolation algorithms, finite element algorithms, and others, the idea of function approximation is also required. According to the function properties in Sobolev spaces, appropriate approximation functions and algorithm strategies are selected to achieve efficient calculation and processing of complex functions. Nikol’skii’s problem is the minimization of the norm of the error functional by coefficients and by nodes such as that in [3]. The minimization of the norm of the error functional with fixed nodes by coefficients is Sard’s problem. The corresponding solutions are called the optimal quadrature formulas in the sense of Nikol’skii and in the sense of Sard, respectively. Reducing errors and studying optimal quadrature formulas can help calculate the loss functions involved in the training and evaluation of machine learning models. This, in turn, can improve the generalization ability and prediction accuracy of the models, and has potential application value in fields such as image recognition and speech recognition.

Work [1] studies Sard’s problem of constructing optimal quadrature formulas in the $W_2^{(m,0)}$ space. So, in this paper, we will only consider the case where $m=7$.

We denote a class of functions $\phi$ by $W_2^{(7,0)}$ defined on the interval $[0,1]$, which possesses an absolutely continuous sixth derivative on $[0,1]$ and whose seventh derivative is in $L_2(0,1)$, under the pseudo-inner product

$$⟨\phi ,\psi ⟩={\int }_0^1\left({\phi }^{(7)}(x)+\phi (x)\right)\left({\psi }^{(7)}(x)+\psi (x)\right)dx.$$

(1)

The class $W_2^{(7,0)}$ is a Hilbert space if we identify functions that differ by a solution of the equation $f^{(7)}(x)+f(x)=0$. The norm

$$∥\phi ∥={\left({\int }_0^1{\left({\phi }^{(7)}(x)+\phi (x)\right)}^{2}dx\right)}^{{\displaystyle \frac{1}{2}}}$$

corresponds to the inner product (1).

In Sard’s problem, for a function $\phi$ from the space $W_2^{(7,0)}$, we consider the following quadrature formula:

$${\int }_0^1\phi (x)dx\cong \sum _{\beta =0}^N{C}_{\beta }\phi (x_{\beta }),$$

(2)

where $C_{\beta }$ are coefficients and $x_{\beta }$ are nodes situated in the interval $[0,1]$, $\phi (x_{\beta })$ are given values, and N is a natural number. The error of the quadrature formula is

$$(l,\phi )={\int }_0^1\phi (x)dx-\sum _{\beta =0}^N{C}_{\beta }\phi (x_{\beta }),$$

(3)

and $(l,\phi )={\int }_{-\infty }^{\infty }l(x)\phi (x)dx$ is the value of the error functional l at the given function $\phi$. The error functional l has the form

$$l(x)={\epsilon }_{[0,1]}(x)-\sum _{\beta =0}^N{C}_{\beta }\delta (x-x_{\beta }),$$

(4)

where ${\epsilon }_{[0,1]}(x)$ is the characteristic function of the interval [0, 1], and $\delta$ is Dirac’s delta-function.

By the Cauchy–Schwarz inequality, the absolute value of the error (3) is estimated by the norm

$${∥l∥}_{W_2^{{(7,0)}^{\ast }}}=\underset{{∥\phi ∥}_{W_2^{(7,0)}}=1}{sup}\mid (l,\phi )\mid$$

(5)

of the error functional l as follows:

$$\mid (l,\phi )\mid \le {∥\phi ∥}_{W_2^{(7,0)}}{∥l∥}_{W_2^{{(7,0)}^{\ast }}},$$

where $W_2^{{(7,0)}^{\ast }}$ is the conjugate space of the space $W_2^{(7,0)}$.

The construction of optimal quadrature formulas in the space $W_2^{(7,0)}$ with fixed nodes and given $x_{\beta }$ is to find such coefficients $C_{\beta }$ to satisfy the equation

$${∥\stackrel{\circ }{l}∥}_{W_2^{{(7,0)}^{\ast }}}=\underset{C_{\beta }}{inf}{∥l∥}_{W_2^{{(7,0)}^{\ast }}}.$$

(6)

In short, we will find the minimum of the norm (5) of the error functional l by coefficients $C_{\beta }$ for fixed $x_{\beta }$.

Thus, if we want to construct the optimal quadrature formulas in the $W_2^{(7,0)}$ space, we have to solve two problems. First, we need to calculate the norm of the error functional l for the given quadrature Formula (2). Second, find such values of the coefficients $C_{\beta }$, which satisfy equality (6).

The rest of the paper is organized as follows: In Section 2, we use the extremal function of the error functional (4) to find the norm of the error functional and give a representation of the norm. In Section 3, we obtain the system of linear equations for coefficients of the optimal quadrature formulas in the space $W_2^{(7,0)}$, and the existence and uniqueness of the solution of this system are to be proved. Section 4 includes the formulas of the coefficients of the optimal quadrature formulas. In this section, we obtain the optimal quadrature formulas in the sense of Sard in the space $W_2^{(7,0)}$. In Section 5, we calculate the explicit coefficients of this problem.

2. The Extremal Function and Representation of the Norm of the Error Functional $\mathit{l}(\mathit{x})$

In order to solve the first problem, we should use the extremal function of the given functional. So, the extremal function of the functional l is the function ${\psi }_l$ satisfying the equation

$$(l,{\psi }_l)={∥l∥}_{W_2^{{(7,0)}^{\ast }}}{∥{\psi }_l∥}_{W_2^{(7,0)}}.$$

Since $W_2^{(7,0)}$ is a Hilbert space, using the Riesz theorem on the general form of a linear continuous functional on a Hilbert space, a unique extremal function ${\psi }_l(x)\in W_2^{(7,0)}$ can be found for the error functional $l\in W_2^{{(7,0)}^{\ast }}$. Then, for the functional $l(x)$ and for any $\phi \in W_2^{(7,0)}$, there exists such a function ${\psi }_l$ satisfying equality

$$(l,\phi )=⟨{\psi }_l,\phi ⟩$$

(7)

and ${∥l∥}_{W_2^{{(7,0)}^{\ast }}}={∥{\psi }_l∥}_{W_2^{(7,0)}}$. Here, $⟨{\psi }_l,\phi ⟩$ is the inner product defined by equality (1) in the space $W_2^{(7,0)}$.

Then, we will find the solution to the equation of equality (7). We consider a situation when $\phi ={\psi }_l$. We have

$$(l,{\psi }_l)={∥l∥}_{W_2^{{(7,0)}^{\ast }}}^2.$$

Integrating by parts the right hand side of (7), we obtain

$$(l,\phi )=-{\int }_0^1\left({\psi }_l^{(14)}(x)-{\psi }_l(x)\right)\phi (x)dx+\sum _{s=0}^6{(-1)}^s\left({\psi }_l^{(7+s)}(x)+{\psi }_l^{(s)}(x)\right){\phi }^{(6-s)}(x){\mid }_0^1.$$

(8)

Taking into account the function ${\psi }_l$ is unique and Equation (8), we can obtain the following form:

$${\psi }_l^{(14)}(x)-{\psi }_l(x)=-l(x)$$

(9)

with the boundary conditions

$$\left({\psi }_l^{(7+s)}(x)+{\psi }_l^{(s)}(x)\right){\mid }_{x=0}^{x=1}=0,s=0,1,\dots ,6.$$

The solution to Equation (9) with the boundary conditions is as follows:

$${\psi }_l(x)=-l(x)\ast G_7(x)+Y(x),$$

(10)

where

$$G_7(x)={\displaystyle \frac{sgn(x)}{14}}\left(sinh(x)+\sum _{n=1}^6{e}^{x\cos ({\displaystyle \frac{\pi n}{7}})}\cos \left(x\sin ({\displaystyle \frac{\pi n}{7}})+{\displaystyle \frac{\pi n}{7}}\right)\right)$$

(11)

and

$$\begin{array}{cc}\hfill Y(x)& =d_0{e}^{-x}+\sum _{k=1}^3{e}^{x\cos {\displaystyle \frac{(2k-1)\pi }{7}}}\left(d_{1,k}\cos \left(x\sin {\displaystyle \frac{(2k-1)\pi }{7}}\right)\right.\hfill \end{array}$$

$$\begin{array}{c}\hfill \left.+d_{2,k}\sin \left(x\sin {\displaystyle \frac{(2k-1)\pi }{7}}\right)\right),\end{array}$$

(12)

where $d_0,d_{1,k}$, and $d_{2,k}$ are constants. We should impose the following conditions:

$$(l,e^{-x})=0,$$

(13)

$$\left(l,e^{x\cos {\displaystyle \frac{(2j-1)\pi }{7}}}\cos \left(x\sin {\displaystyle \frac{(2j-1)\pi }{7}}\right)\right)=0,j=1,2,3,$$

(14)

$$\left(l,e^{x\cos {\displaystyle \frac{(2j-1)\pi }{7}}}\sin \left(x\sin {\displaystyle \frac{(2j-1)\pi }{7}}\right)\right)=0,j=1,2,3.$$

(15)

That means the quadrature formula (2) is exact for linear combinations of functions

$$e^{-x},e^{-x\cos {\displaystyle \frac{2\pi k}{7}}}\cos (x\sin {\displaystyle \frac{2\pi k}{7}})and{e}^{-x\cos {\displaystyle \frac{2\pi k}{7}}}\sin (x\sin {\displaystyle \frac{2\pi k}{7}})$$

for $k=1,2,3$.

Then, using Equations (13)–(15), we obtain

$$(l,Y(x))=0,$$

where Y(x) is the function defined by (12). Using (10), we obtain

$${∥l∥}^2=(l,{\psi }_l)=-{\int }_{-\infty }^{\infty }l(x)[l(x)\ast G_7(x)]dx.$$

After some calculation, we obtain

$$\begin{array}{cc}\hfill {∥l∥}^2& =2\sum _{\beta =0}^N{C}_{\beta }{\int }_0^1{G}_7(x-x_{\beta })dx-\sum _{\beta =0}^N\sum _{\gamma =0}^N{C}_{\beta }{C}_{\gamma }{G}_7(x_{\beta }-x_{\gamma })\hfill \end{array}$$

$$\begin{array}{c}\hfill -{\int }_0^1{\int }_0^1{G}_7(x-y)dxdy.\end{array}$$

(16)

Thus, the first problem is solved.

3. The System of Linear Equations for Coefficients ${\mathit{C}}_{\mathit{\beta }}$

We need to find the minimization of the square of the norm (16) for the error functional l by coefficients $C_{\beta }$. In order to find a conditional minimum point of function (16), we apply the Lagrange method. Then, we define $\mathbf{C}=(C_0,C_1,\dots ,C_N)$ and $\mathbf{d}=(d_0,d_{1,1},d_{1,2},d_{1,3},d_{2,1},d_{2,2},d_{2,3})$. We have the following function:

$$\begin{array}{cc}\hfill \mathsf{\Psi }(\mathbf{C},\mathbf{d})& ={∥l∥}^2-2d_0(l,e^{-x})\hfill \\ & -2\sum _{k=1}^3\left[d_{1,k}\left(l,e^{-x\cos {\displaystyle \frac{2\pi k}{7}}}\cos (x\sin {\displaystyle \frac{2\pi k}{7}})\right)+d_{2,k}\left(l,e^{-x\cos {\displaystyle \frac{2\pi k}{7}}}\sin (x\sin {\displaystyle \frac{2\pi k}{7}})\right)\right]\hfill \end{array},$$

where $d_0,d_{1,1},d_{1,2},d_{1,3},d_{2,1},d_{2,2},\mathrm{and}{d}_{2,3}$ are Lagrange multipliers.

Let the partial derivatives of the function $\mathsf{\Psi }(\mathbf{C},\mathbf{d})$ by coefficients $C_{\beta }$ and by $d_0,d_{1,1},d_{1,2},$$d_{1,3},d_{2,1},d_{2,2}$, and $d_{2,3}$ be equal to zero; we can obtain the following difference system of $N+8$ linear equations with $N+8$ unknowns:

$$\sum _{\gamma =0}^N{C}_{\gamma }{G}_7(x_{\beta }-x_{\gamma })+P(x_{\beta },d_0,d_{1,k},d_{2,k})=f_7(x_{\beta }),\beta =0,1,\dots ,N,$$

(17)

$$\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-x_{\gamma }}=1-{\displaystyle \frac{1}{e}},$$

(18)

$$\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-x_{\gamma }\cos {\displaystyle \frac{2k\pi }{7}}}\cos (x_{\gamma }\sin {\displaystyle \frac{2k\pi }{7}})=g_{1,k},k=1,2,3,$$

(19)

$$\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-x_{\gamma }\cos {\displaystyle \frac{2k\pi }{7}}}\sin (x_{\gamma }\sin {\displaystyle \frac{2k\pi }{7}})=g_{2,k},k=1,2,3,$$

(20)

where

$$\begin{array}{cc}\hfill P(x_{\beta },d_0,d_{1,k},d_{2,k})& =d_0{e}^{-x_{\beta }}\hfill \\ & +\sum _{k=1}^3{e}^{-x_{\beta }\cos {\displaystyle \frac{2k\pi }{7}}}\left(d_{1,k}\cos (x_{\beta }\sin {\displaystyle \frac{2\pi k}{7}})+d_{2,k}\sin (x_{\beta }\sin {\displaystyle \frac{2\pi k}{7}})\right),\hfill \end{array}$$

$$f_7(x_{\beta })={\int }_0^1{G}_7(x-x_{\beta })dx,$$

$$g_{1,k}=\cos {\displaystyle \frac{2k\pi }{7}}-e^{-\cos {\displaystyle \frac{2k\pi }{7}}}\cos (\sin {\displaystyle \frac{2k\pi }{7}}+{\displaystyle \frac{2k\pi }{7}}),k=1,2,3,$$

$$g_{2,k}=\sin {\displaystyle \frac{2k\pi }{7}}-e^{-\cos {\displaystyle \frac{2k\pi }{7}}}\sin (\sin {\displaystyle \frac{2k\pi }{7}}+{\displaystyle \frac{2k\pi }{7}}),k=1,2,3,$$

and $G_7(x)$ is defined by (11).

The solution to this system exists and is unique in the sense of Sard in a Hilbert space. Then, we consider the case of equally spaced nodes. Suppose $x_{\beta }=h\beta ,\beta =0,1,2,\dots ,N,h={\displaystyle \frac{1}{N}},N=1,2,\dots$

We also suppose $C_{\beta }=0$ when $\beta <0$ and $\beta >N$. Then, we use the convolution of two discrete argument functions $\phi (h\beta )$ and $\psi (h\beta )$ and have the following form:

$$\phi (h\beta )\ast \psi (h\beta )=\sum _{\gamma =-\infty }^{\infty }\phi (h\gamma )·\psi (h\beta -h\gamma ).$$

We write the system (17)–(20) in this form

$$C_{\beta }\ast G_7(h\beta )+P(h\beta ,d_0,d_{1,k},d_{2,k})=f_7(h\beta ),\beta =0,1,\dots ,N,$$

(21)

$$\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-h\gamma }=1-{\displaystyle \frac{1}{e}},$$

(22)

$$\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-h\gamma \cos {\displaystyle \frac{2k\pi }{7}}}\cos (h\gamma \sin {\displaystyle \frac{2k\pi }{7}})=g_{1,k},k=1,2,3,$$

(23)

$$\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-h\gamma \cos {\displaystyle \frac{2k\pi }{7}}}\sin (h\gamma \sin {\displaystyle \frac{2k\pi }{7}})=g_{2,k},k=1,2,3,$$

(24)

where some functions are defined by the present work.

4. The Formulas of the Coefficients

In order to solve the system (21)–(24), we need to use the differential operator ${\displaystyle \frac{d^{14}}{d{x}^{14}}}-1$. We use the function $D_7(h\beta )$ of discrete argument which was constructed in [1] and satisfies the equation

$$D_7(h\beta )\ast G_7(h\beta )={\delta }_d(h\beta ),$$

(25)

where

$$G_7(h\beta )={\displaystyle \frac{sgn\left(h\beta \right)}{14}}\left(sinh(h\beta )+\sum _{n=1}^6{e}^{h\beta \cos ({\displaystyle \frac{\pi n}{7}})}\cos \left(h\beta \sin ({\displaystyle \frac{\pi n}{7}})+{\displaystyle \frac{\pi n}{7}}\right)\right)$$

(26)

and ${\delta }_d(h\beta )$ is the discrete delta-function and has the form

$${\delta }_d(h\beta )=\left\{\begin{array}{c}1,\beta =0,\hfill \\ 0,\beta \ne 0.\hfill \end{array}\right.$$

Following [1], we have

Theorem 1.

The discrete analog of the differential operator ${\displaystyle \frac{d^{14}}{d{x}^{14}}}-1$ satisfying Equation (25) has the form

$$D_7(h\beta )={\displaystyle \frac{7}{K}}\left\{\begin{array}{c}{\displaystyle \sum _{n=1}^6{A}_n{\lambda }_n^{\mid \beta \mid -1},\mid \beta \mid \ge 2}\hfill \\ {\displaystyle 1+\sum _{n=1}^6{A}_n,\mid \beta \mid =1,}\hfill \\ {\displaystyle M_1-\frac{K_1}{K}+\sum _{n=1}^6\frac{A_n}{{\lambda }_n},\beta =0}\hfill \end{array}\right.$$

(27)

where $k=1,2,3$ and

$$\begin{array}{cc}\hfill A_n& ={\displaystyle \frac{\left({\lambda }_n^2-2{\lambda }_{n}cosh(h)+1\right)B_{12}({\lambda }_n)}{{\lambda }_n{P}_{12}^{\prime }({\lambda }_n)}},n=1,2,\dots ,6,\hfill \\ \hfill K& =\sum _{k=1}^3{a}_{1,k}+sinh(h),\hfill \\ \hfill K_1& =\sum _{k=1}^3\left(b_{1,k}sinh(h)+a_{2,k}+a_{1,k}\left(\sum _{j=1,j\ne k}^3{b}_{1,j}-2cosh(h)\right)\right),\hfill \\ \hfill M_1& =\sum _{k=1}^3{b}_{1,k}-2cosh(h),\hfill \\ \hfill B_{12}(\lambda )& =\prod _{k=1}^3({\lambda }^4+b_{1,k}{\lambda }^3+b_{2,k}{\lambda }^2+b_{1,k}\lambda +1),\hfill \\ \hfill P_{12}(\lambda )& =\left(sinh(h)+\left({\lambda }^2-2\lambda cosh(h)+1\right)\sum _{j=1}^3{\displaystyle \frac{a_{1,j}{\lambda }^2+a_{2,j}\lambda +a_{1,j}}{{\lambda }^4+b_{1,j}{\lambda }^3+b_{2,k}{\lambda }^2+b_{1,k}\lambda +1}}\right)B_{12}(\lambda ),\hfill \\ \hfill a_{1,k}& =2·\left[\cos {\displaystyle \frac{k\pi }{7}}\cos (h\sin {\displaystyle \frac{k\pi }{7}})sinh(h\cos {\displaystyle \frac{k\pi }{7}})-\sin {\displaystyle \frac{k\pi }{7}}\sin (h\sin {\displaystyle \frac{k\pi }{7}})cosh(h\cos {\displaystyle \frac{k\pi }{7}})\right],\hfill \\ \hfill a_{2,k}& =-2·\left[\cos {\displaystyle \frac{k\pi }{7}}sinh(2h\cos {\displaystyle \frac{k\pi }{7}})-\sin {\displaystyle \frac{k\pi }{7}}cosh(2h\cos {\displaystyle \frac{k\pi }{7}})\right],\hfill \\ \hfill b_{1,k}& =-4·\cos (h\sin {\displaystyle \frac{k\pi }{7}})cosh(h\cos {\displaystyle \frac{k\pi }{7}}),\hfill \\ \hfill b_{2,k}& =2·\left[1+\cos (2h\sin {\displaystyle \frac{k\pi }{7}})+cosh(2h\cos {\displaystyle \frac{k\pi }{7}})\right],\hfill \end{array}$$

${\lambda }_n,n=1,2,\dots ,6$ are roots of the polynomial $P_{12}(\lambda )$ with $\mid {\lambda }_n\mid <1$, and h is a small positive parameter.

Theorem 2.

The discrete analog $D_7(h\beta )$ for $k=1,2,3$ satisfies the following equations:

$$\begin{array}{cc}\hfill & (1)D_7(h\beta )\ast e^{h\beta }=0,\hfill \\ \hfill & (2)D_7(h\beta )\ast e^{-h\beta }=0,\hfill \\ \hfill & (3)D_7(h\beta )\ast e^{h\beta \cos ({\displaystyle \frac{2k\pi }{7}})}\cos (h\beta \sin ({\displaystyle \frac{2k\pi }{7}}))=0,\hfill \\ \hfill & (4)D_7(h\beta )\ast e^{-h\beta \cos ({\displaystyle \frac{2k\pi }{7}})}\cos (h\beta \sin ({\displaystyle \frac{2k\pi }{7}}))=0,\hfill \\ \hfill & (5)D_7(h\beta )\ast e^{h\beta \cos ({\displaystyle \frac{2k\pi }{7}})}\sin (h\beta \sin ({\displaystyle \frac{2k\pi }{7}}))=0,\hfill \\ \hfill & (6)D_7(h\beta )\ast e^{-h\beta \cos ({\displaystyle \frac{2k\pi }{7}})}\sin (h\beta \sin ({\displaystyle \frac{2k\pi }{7}}))=0.\hfill \end{array}$$

Next, we introduce the following functions:

$$v_7(h\beta )=C_{\beta }\ast G_7(h\beta ),$$

$$u_7(h\beta )=v_7(h\beta )+P(h\beta ,d_0,d_{1,k},d_{2,k}).$$

Then, for the coefficients of the optimal quadrature formula, we obtain

$$C_{\beta }=D_7(h\beta )\ast u_7(h\beta ).$$

5. The Calculation About Optimal Coefficients

If we want to obtain the explicit value of the coefficients of the optimal quadrature formula, calculating in the manner of work [2], we have to calculate the convolution $D_7(h\beta )\ast u_7(h\beta )$.

Theorem 3.

The convolution $D_7(h\beta )\ast u_7(h\beta )$ has the form

$$C_{\beta }=\left\{\begin{array}{cc}{\displaystyle D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =0}+a+\sum _{n=1}^6(a_n+{\lambda }_n^N{b}_n),}\hfill & \beta =0,\hfill \\ {\displaystyle D_7(h\beta )\ast f_7(h\beta )+\sum _{n=1}^6(a_n{\lambda }_n^{\beta }+b_n{\lambda }_n^{N-\beta }),}\hfill & \beta =1,2,\dots ,N-1,\hfill \\ {\displaystyle D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =N}+b+\sum _{n=1}^6(a_n{\lambda }_n^N+b_n),}\hfill & \beta =N,\hfill \end{array}\right.$$

(28)

where

$$\begin{array}{cc}\hfill a& ={\displaystyle \frac{7}{K}}\left(-{\displaystyle \frac{1}{14}}Q(-h)+P(-h,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h)\right),\hfill \\ \hfill a_n& ={\displaystyle \frac{A_{n}7}{{\lambda }_{n}K}}\sum _{\gamma =1}^{\infty }{\lambda }_n^{\gamma }\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right),\hfill \\ \hfill b& ={\displaystyle \frac{7}{K}}\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right),\hfill \\ \hfill b_n& ={\displaystyle \frac{A_{n}7}{{\lambda }_{n}K}}\sum _{\gamma =1}^{\infty }{\lambda }_n^{\gamma }\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right).\hfill \end{array}$$

Proof. 

We consider the convolution of two discrete functions and take Equation (27) into account. Then, we have

$$\begin{array}{cc}\hfill C_{\beta }& =D_7(h\beta )\ast u_7(h\beta )\hfill \\ \hfill & =\sum _{\gamma =-\infty }^{\infty }{D}_7(h\beta -h\gamma )u_7(h\gamma )\hfill \\ \hfill & =\sum _{\gamma =1}^{\infty }{D}_7(h\beta +h\gamma )\left(-{\displaystyle \frac{1}{14}}Q(h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})\right)+\sum _{\gamma =0}^N{D}_7(h\beta -h\gamma )f_7(h\gamma )\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7\left(h(\gamma +N)-h\beta \right)\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})\right)\hfill \\ \hfill & =\sum _{\gamma =-\infty }^{\infty }{D}_7(h\beta -h\gamma )f_7(h\gamma )\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7(h\beta +h\gamma )\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7(h(\gamma +N)-h\beta )\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ \hfill & =D_7(h\beta )\ast f_7(h\beta )+\sum _{\gamma =1}^{\infty }{D}_7(h\beta +h\gamma )\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7(h(\gamma +N)-h\beta )\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right).\hfill \end{array}$$

where

$$\begin{array}{cccc}{d}_0^{-}=d_0-b_0& d_{1,k}^{-}=d_{1,k}-b_{1,k}& d_{2,k}^{-}=d_{2,k}-b_{2,k}& ,k=1,2,3\\ d_0^{+}=d_0+b_0& d_{1,k}^{+}=d_{1,k}+b_{1,k}& d_{2,k}^{+}=d_{2,k}+b_{2,k}& ,k=1,2,3.\end{array}$$

When $\beta =0$, we have

$$\begin{array}{cc}\hfill C_0& =D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =0}+\sum _{\gamma =1}^{\infty }{D}_7(h\gamma )\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7(h(\gamma +N))\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ \hfill & =D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =0}+{\displaystyle \frac{7}{K}}\left(-{\displaystyle \frac{1}{14}}Q(-h)+P(-h,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h)\right)\hfill \\ & +{\displaystyle \frac{7}{K}}\sum _{\gamma =1}^{\infty }(\sum _{n=1}^6{A}_n{\lambda }_n^{\gamma -1})\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +{\displaystyle \frac{7}{K}}\sum _{\gamma =1}^{\infty }(\sum _{n=1}^6{A}_n{\lambda }_n^{N+\gamma -1})\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ \hfill & =D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =0}+a+\sum _{n=1}^6(a_n+{\lambda }_n^N{b}_n).\hfill \end{array}$$

For $\beta =1,2,\dots ,N-1$, we have

$$\begin{array}{cc}\hfill C_{\beta }& =D_7(h\beta )\ast f_7(h\beta )\hfill \\ & +{\displaystyle \frac{7}{K}}\sum _{\gamma =1}^{\infty }(\sum _{n=1}^6{A}_n{\lambda }_n^{\beta +\gamma -1})\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +{\displaystyle \frac{7}{K}}\sum _{\gamma =1}^{\infty }(\sum _{n=1}^6{A}_n{\lambda }_n^{N+\gamma -\beta -1})\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ \hfill & =D_7(h\beta )\ast f_7(h\beta )+\sum _{n=1}^6(a_n{\lambda }_n^{\beta }+b_n{\lambda }_n^{N-\beta }),\hfill \end{array}$$

When $\beta =N$, we have

$$\begin{array}{cc}\hfill C_N& =D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =N}\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7(h\gamma +1)\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +\sum _{\gamma =1}^{\infty }{D}_7(h\gamma )\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ \hfill & =D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =N}+{\displaystyle \frac{7}{K}}\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ & +{\displaystyle \frac{7}{K}}\sum _{\gamma =1}^{\infty }(\sum _{n=1}^6{A}_n{\lambda }_n^{\gamma +N-1})\left(-{\displaystyle \frac{1}{14}}Q(-h\gamma )+P(-h\gamma ,d_0^{-},d_{1,k}^{-},d_{2,k}^{-})-f_7(-h\gamma )\right)\hfill \\ & +{\displaystyle \frac{7}{K}}\sum _{\gamma =1}^{\infty }(\sum _{n=1}^6{A}_n{\lambda }_n^{\gamma -1})\left({\displaystyle \frac{1}{14}}Q(h\gamma +1)+P(h\gamma +1,d_0^{+},d_{1,k}^{+},d_{2,k}^{+})-f_7(h\gamma +1)\right)\hfill \\ \hfill & =D_7(h\beta )\ast f_7(h\beta ){\mid }_{\beta =N}+b+\sum _{n=1}^6(a_n{\lambda }_n^N+b_n).\hfill \end{array}$$

Theorem 3 is proved. □

It is easy to see that if we want to solve this problem, we will consider the convolution $D_7(h\beta )\ast f_7(h\beta )$. And in (28), there are 14 unknowns, and we also need to find their solutions. Next, we will find the expression for $f_7(h\beta )$. In the present work, we obtain

$$f_7(h\beta )={\int }_0^1{G}_7(x-h\beta )dx,$$

where $G_7(x)$ is defined by (26). After some calculations as work [4], we can obtain $f_7(h\beta )$, which has following form:

$$\begin{array}{cc}\hfill f_7(h\beta )& ={\displaystyle \frac{1}{28}}\left[e^{-h\beta }(e+1)+e^{h\beta }(e^{-1}+1)+2\sum _{k=1}^6{e}^{-h\beta \cos {\displaystyle \frac{k\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{k\pi }{7}})\right.\hfill \\ & +2\sum _{k=1}^6{e}^{\cos {\displaystyle \frac{k\pi }{7}}}\cos (\sin {\displaystyle \frac{k\pi }{7}})e^{-h\beta \cos {\displaystyle \frac{k\pi }{7}}}\cos \left(\sin ({\displaystyle \frac{k\pi }{7}}h\beta )\right)\hfill \\ & \left.+2\sum _{k=1}^6{e}^{\cos {\displaystyle \frac{k\pi }{7}}}\sin (\sin {\displaystyle \frac{k\pi }{7}})e^{-h\beta \cos {\displaystyle \frac{k\pi }{7}}}\sin \left(\sin ({\displaystyle \frac{k\pi }{7}}h\beta )\right)\right]-1.\hfill \end{array}$$

Next, taking Theorem 2 into account, we consider the convolution $D_7(h\beta )\ast f_7(h\beta )$. We have

$$\begin{array}{cc}\hfill D_7(h\beta )\ast f_7(h\beta )& =D_7(h\beta )\ast (-1)\hfill \\ & =-\sum _{-\infty }^{\infty }D(h\gamma )\hfill \\ & =-\left(D(0)+2D(h)+2\sum _{\gamma =2}^{\infty }D(h\gamma )\right).\hfill \end{array}$$

Then, we can obtain the value of the convolution as the following form:

$$D_7(h\beta )\ast f_7(h\beta )=T,$$

where

$$T=-{\displaystyle \frac{7}{K}}(M_1-{\displaystyle \frac{K_1}{K}}+\sum _{k=1}^6{\displaystyle \frac{A_k}{{\lambda }_k}}+2\sum _{k=1}^6{A}_k+2\sum _{\gamma =2}^{\infty }\sum _{k=1}^6{A}_k{\lambda }_k^{\gamma -1}+2).$$

So, if we consider Theorem 3, we can obtain a simple form of the expression of $C_{\beta }$ in the sense of Sard in the space $W_2^{(7,0)}$. The simple form is as follows:

$$C_{\beta }=\left\{\begin{array}{cc}{\displaystyle T+a+\sum _{n=1}^6(a_n+{\lambda }_n^N{b}_n),}\hfill & \beta =0,\hfill \\ {\displaystyle T+\sum _{n=1}^6(a_n{\lambda }_n^{\beta }+b_n{\lambda }_n^{N-\beta }),}\hfill & \beta =1,2,\dots ,N-1,\hfill \\ {\displaystyle T+b+\sum _{n=1}^6(a_n{\lambda }_n^N+b_n),}\hfill & \beta =N.\hfill \end{array}\right.$$

In this form, we have 14 unknowns. To find out about them, we will take Equations (21)–(24). In the rest of the paper, we will first calculate the convolution $G(h\beta )\ast C_{\beta }$. Then, use Equations (22) and (23) and the equality of the coefficients on the right and left sides of Equation (21) to construct a system of linear equations which includes 21 equations and 21 unknowns.

Next, we calculate the convolution $G(h\beta )\ast C_{\beta }$.

$$\begin{array}{cc}\hfill G(h\beta )\ast C_{\beta }& =\sum _{\gamma =0}^N{C}_{\gamma }G(h\beta -h\gamma )\hfill \\ \hfill & =\sum _{\gamma =0}^{\beta }{C}_{\gamma }{\displaystyle \frac{1}{14}}\left(sinh(h\beta -h\gamma )+\sum _{k=1}^6{e}^{(h\beta -h\gamma )\cos {\displaystyle \frac{k\pi }{7}}}\cos ((h\beta -h\gamma )\sin {\displaystyle \frac{k\pi }{7}}+{\displaystyle \frac{k\pi }{7}})\right)\hfill \\ & -\sum _{\gamma =\beta +1}^N{C}_{\gamma }{\displaystyle \frac{1}{14}}\left(sinh(h\beta -h\gamma )+\sum _{k=1}^6{e}^{(h\beta -h\gamma )\cos {\displaystyle \frac{k\pi }{7}}}\cos ((h\beta -h\gamma )\sin {\displaystyle \frac{k\pi }{7}}+{\displaystyle \frac{k\pi }{7}})\right)\hfill \\ \hfill & =-\sum _{\gamma =0}^N{C}_{\gamma }{\displaystyle \frac{1}{14}}\left(sinh(h\beta -h\gamma )+\sum _{k=1}^6{e}^{(h\beta -h\gamma )\cos {\displaystyle \frac{k\pi }{7}}}\cos ((h\beta -h\gamma )\sin {\displaystyle \frac{k\pi }{7}}+{\displaystyle \frac{k\pi }{7}})\right)\hfill \\ \hfill & +\sum _{\gamma =0}^{\beta }{C}_{\gamma }{\displaystyle \frac{1}{7}}\left(sinh(h\beta -h\gamma )+\sum _{k=1}^6{e}^{(h\beta -h\gamma )\cos {\displaystyle \frac{k\pi }{7}}}\cos ((h\beta -h\gamma )\sin {\displaystyle \frac{k\pi }{7}}+{\displaystyle \frac{k\pi }{7}})\right)\hfill \\ \hfill & =S_1-S_2,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill S_1& =\sum _{\gamma =0}^{\beta }{C}_{\gamma }{\displaystyle \frac{1}{7}}\left(sinh(h\beta -h\gamma )+\sum _{k=1}^6{e}^{(h\beta -h\gamma )\cos {\displaystyle \frac{k\pi }{7}}}\cos ((h\beta -h\gamma )\sin {\displaystyle \frac{k\pi }{7}}+{\displaystyle \frac{k\pi }{7}})\right),\hfill \\ \hfill S_2& =\sum _{\gamma =0}^N{C}_{\gamma }{\displaystyle \frac{1}{14}}\left(sinh(h\beta -h\gamma )+\sum _{k=1}^6{e}^{(h\beta -h\gamma )\cos {\displaystyle \frac{k\pi }{7}}}\cos ((h\beta -h\gamma )\sin {\displaystyle \frac{k\pi }{7}}+{\displaystyle \frac{k\pi }{7}})\right).\hfill \end{array}$$

After some calculation, we have

$$\begin{array}{cc}\hfill S_1& =e^{h\beta }·{\displaystyle \frac{1}{14}}\left(T{\displaystyle \frac{e^h}{e^h-1}}+a+\sum _{k=1}^6(a_n{\displaystyle \frac{e^h}{e^h-{\lambda }_n}}+b_n{\lambda }^N{\displaystyle \frac{{\lambda }_n{e}^h}{{\lambda }_n{e}^h-1}})\right)\hfill \\ & -e^{-h\beta }·{\displaystyle \frac{1}{14}}\left({\displaystyle \frac{T}{1-e^h}}+a+\sum _{k=1}^6(a_n{\displaystyle \frac{1}{1-{\lambda }_n{e}^h}}+b_n{\displaystyle \frac{{\lambda }_n^{N+1}}{{\lambda }_n-e^h}})\right)\hfill \\ & +{\displaystyle \frac{1}{7}}\sum _{k=1}^6{e}^{(h\beta )\cos {\displaystyle \frac{k\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{k\pi }{7}})\left[a·\cos {\displaystyle \frac{k\pi }{7}}+T{\displaystyle \frac{\cos \frac{k\pi }{7}-e^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7}+\frac{k\pi }{7})}{1+e^{-2h\cos \frac{k\pi }{7}}-2e^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right.\hfill \\ & +\sum _{n=1}^6\left(a_n{\displaystyle \frac{\cos \frac{k\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7}+\frac{k\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{k\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right.\hfill \\ & \left.\left.+b_n{\lambda }_n^N{\displaystyle \frac{\cos \frac{k\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7}+\frac{k\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{k\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right)\right]\hfill \\ & -{\displaystyle \frac{1}{7}}\sum _{k=1}^6{e}^{(h\beta )\cos {\displaystyle \frac{k\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{k\pi }{7}})\left[a·\sin {\displaystyle \frac{k\pi }{7}}+T{\displaystyle \frac{\sin \frac{k\pi }{7}-e^{-h\cos \frac{k\pi }{7}}\sin (h\sin \frac{k\pi }{7}+\frac{k\pi }{7})}{1+e^{-2h\cos \frac{k\pi }{7}}-2e^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right.\hfill \\ & +\sum _{n=1}^6\left(a_n{\displaystyle \frac{\sin \frac{k\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{k\pi }{7}}\sin (h\sin \frac{k\pi }{7}+\frac{k\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{k\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right.\hfill \\ & \left.\left.+b_n{\lambda }_n^N{\displaystyle \frac{\sin \frac{k\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{k\pi }{7}}\sin (h\sin \frac{k\pi }{7}+\frac{k\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{k\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right)\right]+{\displaystyle \frac{1}{14}}\left(T{\displaystyle \frac{e^h+1}{1-e^h}}\right.\hfill \\ & \left.+\sum _{n=1}^6({\displaystyle \frac{a_n{\lambda }_n^{\beta +1}{e}^h+b_n{\lambda }_n^{N-\beta }}{1-{\lambda }_n{e}^h}}+{\displaystyle \frac{b_n{\lambda }_n^{N-\beta }{e}^h+{\lambda }_n^{\beta +1}}{{\lambda }_n-e^h}})\right)\hfill \\ & +{\displaystyle \frac{1}{7}}\sum _{k=1}^6\left[T{\displaystyle \frac{e^{-2h\cos \frac{k\pi }{7}}\cos \frac{k\pi }{7}-e^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7}-\frac{k\pi }{7})}{1+e^{-2h\cos \frac{k\pi }{7}}-2e^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right.\hfill \\ & +\sum _{n=1}^6\left(a_n{\displaystyle \frac{{\lambda }_n^{\beta +2}{e}^{-2h\cos \frac{k\pi }{7}}\cos (\frac{k\pi }{7})-{\lambda }_n^{\beta +1}{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7}-\frac{k\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{k\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right.\hfill \\ & \left.\left.+b_n{\lambda }_n^N{\displaystyle \frac{{\lambda }_n^{-\beta -2}{e}^{-2h\cos \frac{k\pi }{7}}\cos (\frac{k\pi }{7})-{\lambda }_n^{-\beta -1}{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7}-\frac{k\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{k\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{k\pi }{7}}\cos (h\sin \frac{k\pi }{7})}}\right)\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill S_2& =e^{h\beta }·{\displaystyle \frac{1}{28}}(1-{\displaystyle \frac{1}{e}})-e^{-h\beta }·{\displaystyle \frac{1}{28}}(a+eb+T{\displaystyle \frac{e^{h+1}-1}{e^h-1}}+\sum _{n=1}^6(a_n{\displaystyle \frac{1-e^{h+1}{\lambda }_n^{N+1}}{1-{\lambda }_n{e}^h}}+b_n{\displaystyle \frac{{\lambda }_n^{N+1}-e^{h+1}}{{\lambda }_n-e^h}}))\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{\pi }{7}})(\cos {\displaystyle \frac{\pi }{7}}·F_{11}+\sin {\displaystyle \frac{\pi }{7}}·F_{21})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{\pi }{7}})(\cos {\displaystyle \frac{\pi }{7}}·F_{21}-\sin {\displaystyle \frac{\pi }{7}}·F_{11})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{2\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{2\pi }{7}})\left(1-e^{-\cos {\displaystyle \frac{2\pi }{7}}}\cos (\sin {\displaystyle \frac{2\pi }{7}})\right)\hfill \\ & -{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{2\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{2\pi }{7}})e^{-\cos {\displaystyle \frac{2\pi }{7}}}\sin (\sin {\displaystyle \frac{2\pi }{7}})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{3\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{3\pi }{7}})(\cos {\displaystyle \frac{3\pi }{7}}·F_{13}+\sin {\displaystyle \frac{3\pi }{7}}·F_{23})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{3\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{3\pi }{7}})(\cos {\displaystyle \frac{3\pi }{7}}·F_{23}-\sin {\displaystyle \frac{3\pi }{7}}·F_{13})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{4\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{4\pi }{7}})\left(1-e^{-\cos {\displaystyle \frac{4\pi }{7}}}\cos (\sin {\displaystyle \frac{4\pi }{7}})\right)\hfill \\ & -{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{4\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{4\pi }{7}})e^{-\cos {\displaystyle \frac{4\pi }{7}}}\sin (\sin {\displaystyle \frac{4\pi }{7}})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{5\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{5\pi }{7}})(\cos {\displaystyle \frac{5\pi }{7}}·F_{15}+\sin {\displaystyle \frac{5\pi }{7}}·F_{25})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{5\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{5\pi }{7}})(\cos {\displaystyle \frac{5\pi }{7}}·F_{25}-\sin {\displaystyle \frac{5\pi }{7}}·F_{15})\hfill \\ & +{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{6\pi }{7}}}\cos (h\beta \sin {\displaystyle \frac{6\pi }{7}})\left(1-e^{-\cos {\displaystyle \frac{6\pi }{7}}}\cos (\sin {\displaystyle \frac{6\pi }{7}})\right)\hfill \\ & -{\displaystyle \frac{1}{14}}{e}^{h\beta \cos {\displaystyle \frac{6\pi }{7}}}\sin (h\beta \sin {\displaystyle \frac{6\pi }{7}})e^{-\cos {\displaystyle \frac{6\pi }{7}}}\sin (\sin {\displaystyle \frac{6\pi }{7}}),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill F_{1k}& =\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-h\gamma \cos {\displaystyle \frac{k\pi }{7}}}\cos (h\gamma \sin {\displaystyle \frac{k\pi }{7}}),\hfill \\ \hfill F_{2k}& =\sum _{\gamma =0}^N{C}_{\gamma }{e}^{-h\gamma \cos {\displaystyle \frac{k\pi }{7}}}\sin (h\gamma \sin {\displaystyle \frac{k\pi }{7}}).\hfill \end{array}$$

Then, taking Equation (21) into account, we obtain

$$\begin{array}{cc}& S_1-S_2+d_0{e}^{-h\beta }+e^{-h\beta \cos {\displaystyle \frac{2\pi }{7}}}(d_{11}\cos (h\beta \sin {\displaystyle \frac{2\pi }{7}})+d_{21}\sin (h\beta \sin {\displaystyle \frac{2\pi }{7}}))+e^{-h\beta \cos {\displaystyle \frac{4\pi }{7}}}(d_{12}\cos (h\beta \sin {\displaystyle \frac{4\pi }{7}})\hfill \\ & +d_{22}\sin (h\beta \sin {\displaystyle \frac{4\pi }{7}}))+e^{-h\beta \cos {\displaystyle \frac{6\pi }{7}}}(d_{13}\cos (h\beta \sin {\displaystyle \frac{6\pi }{7}})+d_{23}\sin (h\beta \sin {\displaystyle \frac{6\pi }{7}}))=f_7(h\beta ),\hfill \end{array}$$

where $d_0,d_{11},d_{21},d_{12},d_{22},d_{13}$, and $d_{23}$ are unknowns. We can construct a system of linear equations which include 21 equations and 21 unknowns by this equation and (22)–(24). The system of linear equations has the following form:

$$\begin{array}{cc}\hfill a+\sum _{n=1}^6{a}_n{H}_{1,1n}+\sum _{n=1}^6{b}_n{H}_{1,2n}& =T_1,\hfill \\ \hfill -2a+eb+\sum _{n=1}^6{a}_n{H}_{2,1n}+\sum _{n=1}^6{b}_n{H}_{2,2n}+28d_0& =T_2,\hfill \\ \hfill a·\cos {\displaystyle \frac{\pi }{7}}-b{e}^{-\cos {\displaystyle \frac{\pi }{7}}}\cos (\sin {\displaystyle \frac{\pi }{7}}-{\displaystyle \frac{\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{3,1n}+\sum _{n=1}^6{b}_n{H}_{3,2n}+d_{13}& =T_3,\hfill \\ \hfill 2a·\cos {\displaystyle \frac{\pi }{7}}+\sum _{n=1}^6{a}_n{H}_{4,1n}+\sum _{n=1}^6{b}_n{H}_{4,2n}& =T_4,\hfill \\ \hfill a·\cos {\displaystyle \frac{3\pi }{7}}-b{e}^{-\cos {\displaystyle \frac{3\pi }{7}}}\cos (\sin {\displaystyle \frac{3\pi }{7}}-{\displaystyle \frac{3\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{5,1n}+\sum _{n=1}^6{b}_n{H}_{5,2n}+d_{12}& =T_5,\hfill \\ \hfill 2a·\cos {\displaystyle \frac{4\pi }{7}}+\sum _{n=1}^6{a}_n{H}_{6,1n}+\sum _{n=1}^6{b}_n{H}_{6,2n}& =T_6,\hfill \\ \hfill a·\cos {\displaystyle \frac{5\pi }{7}}-b{e}^{-\cos {\displaystyle \frac{5\pi }{7}}}\cos (\sin {\displaystyle \frac{5\pi }{7}}-{\displaystyle \frac{5\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{7,1n}+\sum _{n=1}^6{b}_n{H}_{7,2n}+d_{11}& =T_7,\hfill \\ \hfill 2a·\cos {\displaystyle \frac{6\pi }{7}}+\sum _{n=1}^6{a}_n{H}_{8,1n}+\sum _{n=1}^6{b}_n{H}_{8,2n}& =T_8,\hfill \\ \hfill a·\sin {\displaystyle \frac{\pi }{7}}+b·e^{-\cos {\displaystyle \frac{\pi }{7}}}\sin (\sin {\displaystyle \frac{\pi }{7}}-{\displaystyle \frac{\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{9,1n}+\sum _{n=1}^6{b}_n{H}_{9,2n}+d_{23}& =T_9,\hfill \\ \hfill a·\sin {\displaystyle \frac{2\pi }{7}}+\sum _{n=1}^6{a}_n{H}_{10,1n}+\sum _{n=1}^6{b}_n{H}_{10,2n}& =T_{10},\hfill \\ \hfill a·\sin {\displaystyle \frac{\pi }{7}}+b·e^{-\cos {\displaystyle \frac{3\pi }{7}}}\sin (\sin {\displaystyle \frac{3\pi }{7}}-{\displaystyle \frac{3\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{11,1n}+\sum _{n=1}^6{b}_n{H}_{11,2n}+d_{22}& =T_{11},\hfill \\ \hfill a·\sin {\displaystyle \frac{4\pi }{7}}+\sum _{n=1}^6{a}_n{H}_{12,1n}+\sum _{n=1}^6{b}_n{H}_{12,2n}& =T_{12},\hfill \\ \hfill a·\sin {\displaystyle \frac{5\pi }{7}}+b·e^{-\cos {\displaystyle \frac{5\pi }{7}}}\sin (\sin {\displaystyle \frac{5\pi }{7}}-{\displaystyle \frac{5\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{13,1n}+\sum _{n=1}^6{b}_n{H}_{13,2n}+d_{21}& =T_{13},\hfill \\ \hfill a·\sin {\displaystyle \frac{6\pi }{7}}+\sum _{n=1}^6{a}_n{H}_{14,1n}+\sum _{n=1}^6{b}_n{H}_{14,2n}& =T_{14},\hfill \\ \hfill a+b·{\displaystyle \frac{1}{e}}+\sum _{n=1}^6{a}_n{H}_{15,1n}+\sum _{n=1}^6{b}_n{H}_{15,2n}& =T_{15},\hfill \\ \hfill a+b·e^{-\cos {\displaystyle \frac{2\pi }{7}}}\cos (\sin {\displaystyle \frac{2\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{16,1n}+\sum _{n=1}^6{b}_n{H}_{16,2n}& =T_{16},\hfill \\ \hfill a+b·e^{-\cos {\displaystyle \frac{4\pi }{7}}}\cos (\sin {\displaystyle \frac{4\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{17,1n}+\sum _{n=1}^6{b}_n{H}_{17,2n}& =T_{17},\hfill \\ \hfill a+b·e^{-\cos {\displaystyle \frac{6\pi }{7}}}\cos (\sin {\displaystyle \frac{6\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{18,1n}+\sum _{n=1}^6{b}_n{H}_{18,2n}& =T_{18},\hfill \\ \hfill b·e^{-\cos {\displaystyle \frac{2\pi }{7}}}\sin (\sin {\displaystyle \frac{2\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{19,1n}+\sum _{n=1}^6{b}_n{H}_{19,2n}& =T_{19},\hfill \\ \hfill b·e^{-\cos {\displaystyle \frac{4\pi }{7}}}\sin (\sin {\displaystyle \frac{4\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{20,1n}+\sum _{n=1}^6{b}_n{H}_{20,2n}& =T_{20},\hfill \\ \hfill b·e^{-\cos {\displaystyle \frac{6\pi }{7}}}\sin (\sin {\displaystyle \frac{6\pi }{7}})+\sum _{n=1}^6{a}_n{H}_{21,1n}+\sum _{n=1}^6{b}_n{H}_{21,2n}& =T_{21},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill H_{1,1n}& ={\displaystyle \frac{e^h}{e^h-{\lambda }_n}},\hfill \\ \hfill H_{2,1n}& =-{\displaystyle \frac{{\lambda }_n^{N+1}{e}^{h+1}+1}{1-{\lambda }_n{e}^h}},\hfill \\ \hfill H_{3,1n}& ={\displaystyle \frac{\begin{array}{c}\cos \frac{\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7}+\frac{\pi }{7})-{\lambda }_n^{N+2}{e}^{-2h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\cos (\sin \frac{\pi }{7}-\frac{\pi }{7})\\ +{\lambda }_n^{N+1}{e}^{-h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\cos (\sin \frac{\pi }{7}+h\sin \frac{\pi }{7}-\frac{\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7})}},\hfill \\ \hfill H_{4,1n}& =2{\displaystyle \frac{\cos \frac{2\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7}+\frac{2\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{5,1n}& ={\displaystyle \frac{\begin{array}{c}\cos \frac{3\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7}+\frac{3\pi }{7})-{\lambda }_n^{N+2}{e}^{-2h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\cos (\sin \frac{3\pi }{7}-\frac{3\pi }{7})\\ +{\lambda }_n^{N+1}{e}^{-h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\cos (\sin \frac{3\pi }{7}+h\sin \frac{3\pi }{7}-\frac{3\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{3\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7})}},\hfill \\ \hfill H_{6,1n}& =2{\displaystyle \frac{\cos \frac{4\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7}+\frac{4\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{7,1n}& ={\displaystyle \frac{\begin{array}{c}\cos \frac{5\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7}+\frac{5\pi }{7})-{\lambda }_n^{N+2}{e}^{-2h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\cos (\sin \frac{5\pi }{7}-\frac{5\pi }{7})\\ +{\lambda }_n^{N+1}{e}^{-h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\cos (\sin \frac{5\pi }{7}+h\sin \frac{5\pi }{7}-\frac{5\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{5\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7})}},\hfill \\ \hfill H_{8,1n}& =2{\displaystyle \frac{\cos \frac{6\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7}+\frac{6\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill H_{9,1n}& ={\displaystyle \frac{\begin{array}{c}\sin \frac{\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{\pi }{7}}\sin (h\sin \frac{\pi }{7}+\frac{\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\sin (\sin \frac{\pi }{7}-\frac{\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\sin (\sin \frac{\pi }{7}+h\sin \frac{\pi }{7}-\frac{\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7})}},\hfill \\ \hfill H_{10,1n}& ={\displaystyle \frac{\sin \frac{2\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\sin (h\sin \frac{2\pi }{7}+\frac{2\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{11,1n}& ={\displaystyle \frac{\begin{array}{c}\sin \frac{3\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{3\pi }{7}}\sin (h\sin \frac{3\pi }{7}+\frac{3\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\sin (\sin \frac{3\pi }{7}-\frac{3\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\sin (\sin \frac{3\pi }{7}+h\sin \frac{3\pi }{7}-\frac{3\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{3\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7})}},\hfill \\ \hfill H_{12,1n}& ={\displaystyle \frac{\sin \frac{4\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\sin (h\sin \frac{4\pi }{7}+\frac{4\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{13,1n}& ={\displaystyle \frac{\begin{array}{c}\sin \frac{5\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{5\pi }{7}}\sin (h\sin \frac{5\pi }{7}+\frac{5\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\sin (\sin \frac{5\pi }{7}-\frac{5\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\sin (\sin \frac{5\pi }{7}+h\sin \frac{5\pi }{7}-\frac{5\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{5\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7})}},\hfill \\ \hfill H_{14,1n}& ={\displaystyle \frac{\sin \frac{6\pi }{7}-{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\sin (h\sin \frac{6\pi }{7}+\frac{6\pi }{7})}{1+{\lambda }_n^2{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill H_{15,1n}& ={\displaystyle \frac{e^{h+1}-{\lambda }_n^{N+1}}{e^{h+1}-{\lambda }_{n}e}},\hfill \\ \hfill H_{16,1n}& ={\displaystyle \frac{\begin{array}{c}1-{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\cos (\sin \frac{2\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\cos (\sin \frac{2\pi }{7}+h\sin \frac{2\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{17,1n}& ={\displaystyle \frac{\begin{array}{c}1-{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\cos (\sin \frac{4\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\cos (\sin \frac{4\pi }{7}+h\sin \frac{4\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{18,1n}& ={\displaystyle \frac{\begin{array}{c}1-{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\cos (\sin \frac{6\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\cos (\sin \frac{6\pi }{7}+h\sin \frac{6\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill H_{19,1n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\sin (\sin \frac{2\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\sin (\sin \frac{2\pi }{7}+h\sin \frac{2\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{20,1n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\sin (\sin \frac{4\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\sin (\sin \frac{4\pi }{7}+h\sin \frac{4\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{21,1n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})+{\lambda }_n^{N+2}{e}^{-2h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\sin (\sin \frac{6\pi }{7})\\ -{\lambda }_n^{N+1}{e}^{-h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\sin (\sin \frac{6\pi }{7}+h\sin \frac{6\pi }{7})\end{array}}{1+{\lambda }_n^2{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill H_{1,2n}& ={\displaystyle \frac{{\lambda }_n^{N+1}{e}^h}{{\lambda }_n{e}^h-1}},\hfill \\ \hfill H_{2,2n}& =-{\displaystyle \frac{{\lambda }_n^{N+1}+e^{h+1}}{{\lambda }_n-e^h}},\hfill \\ \hfill H_{3,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N\cos \frac{\pi }{7}-{\lambda }_n^{N-1}{e}^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7}+\frac{\pi }{7})-{\lambda }_n^{-2}{e}^{-2h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\cos (\sin \frac{\pi }{7}-\frac{\pi }{7})\\ +{\lambda }_n^{-1}{e}^{-h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\cos (\sin \frac{\pi }{7}+h\sin \frac{\pi }{7}-\frac{\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7})}},\hfill \\ \hfill H_{4,2n}& =2{\lambda }_n^N{\displaystyle \frac{\cos \frac{2\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7}+\frac{2\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{5,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N\cos \frac{3\pi }{7}-{\lambda }_n^{N-1}{e}^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7}+\frac{3\pi }{7})-{\lambda }_n^{-2}{e}^{-2h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\cos (\sin \frac{3\pi }{7}-\frac{3\pi }{7})\\ +{\lambda }_n^{-1}{e}^{-h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\cos (\sin \frac{3\pi }{7}+h\sin \frac{3\pi }{7}-\frac{3\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{3\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7})}},\hfill \\ \hfill H_{6,2n}& =2{\lambda }_n^N{\displaystyle \frac{\cos \frac{4\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7}+\frac{4\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{7,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N\cos \frac{5\pi }{7}-{\lambda }_n^{N-1}{e}^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7}+\frac{5\pi }{7})-{\lambda }_n^{-2}{e}^{-2h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\cos (\sin \frac{5\pi }{7}-\frac{5\pi }{7})\\ +{\lambda }_n^{-1}{e}^{-h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\cos (\sin \frac{5\pi }{7}+h\sin \frac{5\pi }{7}-\frac{5\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{5\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7})}},\hfill \\ \hfill H_{8,2n}& =2{\lambda }_n^N{\displaystyle \frac{\cos \frac{6\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7}+\frac{6\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill H_{9,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N\sin \frac{\pi }{7}-{\lambda }_n^{N-1}{e}^{-h\cos \frac{\pi }{7}}\sin (h\sin \frac{\pi }{7}+\frac{\pi }{7})+{\lambda }_n^{N-2}{e}^{-2h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\sin (\sin \frac{\pi }{7}-\frac{\pi }{7})\\ -{\lambda }_n^{N-1}{e}^{-h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\sin (\sin \frac{\pi }{7}+h\sin \frac{\pi }{7}-\frac{\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7})}},\hfill \\ \hfill H_{10,2n}& ={\lambda }_n^N{\displaystyle \frac{\sin \frac{2\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}}\sin (h\sin \frac{2\pi }{7}+\frac{2\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{11,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N\sin \frac{3\pi }{7}-{\lambda }_n^{N-1}{e}^{-h\cos \frac{3\pi }{7}}\sin (h\sin \frac{3\pi }{7}+\frac{3\pi }{7})+{\lambda }_n^{N-2}{e}^{-2h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\sin (\sin \frac{3\pi }{7}-\frac{3\pi }{7})\\ -{\lambda }_n^{N-1}{e}^{-h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\sin (\sin \frac{3\pi }{7}+h\sin \frac{3\pi }{7}-\frac{3\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{3\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7})}},\hfill \\ \hfill H_{12,2n}& ={\lambda }_n^N{\displaystyle \frac{\sin \frac{4\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}}\sin (h\sin \frac{4\pi }{7}+\frac{4\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{13,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N\sin \frac{5\pi }{7}-{\lambda }_n^{N-1}{e}^{-h\cos \frac{5\pi }{7}}\sin (h\sin \frac{5\pi }{7}+\frac{5\pi }{7})+{\lambda }_n^{N-2}{e}^{-2h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\sin (\sin \frac{5\pi }{7}-\frac{5\pi }{7})\\ -{\lambda }_n^{N-1}{e}^{-h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\sin (\sin \frac{5\pi }{7}+h\sin \frac{5\pi }{7}-\frac{5\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{5\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7})}},\hfill \\ \hfill H_{14,2n}& ={\lambda }_n^N{\displaystyle \frac{\sin \frac{6\pi }{7}-{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}}\sin (h\sin \frac{6\pi }{7}+\frac{6\pi }{7})}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill H_{15,2n}& ={\displaystyle \frac{e^{h+1}{\lambda }_n^{N+1}-1}{{\lambda }_n{e}^{h+1}-e}},\hfill \\ \hfill H_{16,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N-{\lambda }_n^{N-1}{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})+{\lambda }_n^{-2}{e}^{-2h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\cos (\sin \frac{2\pi }{7})\\ -{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\cos (\sin \frac{2\pi }{7}+h\sin \frac{2\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{17,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N-{\lambda }_n^{N-1}{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})+{\lambda }_n^{-2}{e}^{-2h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\cos (\sin \frac{4\pi }{7})\\ -{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\cos (\sin \frac{4\pi }{7}+h\sin \frac{4\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{18,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^N-{\lambda }_n^{N-1}{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})+{\lambda }_n^{-2}{e}^{-2h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\cos (\sin \frac{6\pi }{7})\\ -{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\cos (\sin \frac{6\pi }{7}+h\sin \frac{6\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill H_{19,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^{N-1}{e}^{-h\cos \frac{2\pi }{7}}\sin (h\sin \frac{2\pi }{7})+{\lambda }_n^{-2}{e}^{-2h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\sin (\sin \frac{2\pi }{7})\\ -{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\sin (\sin \frac{2\pi }{7}+h\sin \frac{2\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{2\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill H_{20,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^{N-1}{e}^{-h\cos \frac{4\pi }{7}}\sin (h\sin \frac{4\pi }{7})+{\lambda }_n^{-2}{e}^{-2h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\sin (\sin \frac{4\pi }{7})\\ -{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\sin (\sin \frac{4\pi }{7}+h\sin \frac{4\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{4\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill H_{21,2n}& ={\displaystyle \frac{\begin{array}{c}{\lambda }_n^{N-1}{e}^{-h\cos \frac{6\pi }{7}}\sin (h\sin \frac{6\pi }{7})+{\lambda }_n^{-2}{e}^{-2h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\sin (\sin \frac{6\pi }{7})\\ -{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\sin (\sin \frac{6\pi }{7}+h\sin \frac{6\pi }{7})\end{array}}{1+{\lambda }_n^{-2}{e}^{-2h\cos \frac{6\pi }{7}}-2{\lambda }_n^{-1}{e}^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill T_1& =1-T{\displaystyle \frac{e^h}{e^h-1}},\hfill \\ \hfill T_2& =1+e-T{\displaystyle \frac{e^{h+1}+1}{e^h-1}},\hfill \\ \hfill T_3& =e^{-\cos {\displaystyle \frac{\pi }{7}}}\cos (\sin {\displaystyle \frac{\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}\cos \frac{\pi }{7}-e^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7}+\frac{\pi }{7})-e^{-2h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\cos (\sin \frac{\pi }{7}-\frac{\pi }{7})\\ +e^{-h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\cos (\sin \frac{\pi }{7}+h\sin \frac{\pi }{7}-\frac{\pi }{7})\end{array}}{1+e^{-2h\cos \frac{\pi }{7}}-2e^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7})}},\hfill \\ \hfill T_4& =1-2T{\displaystyle \frac{\cos \frac{2\pi }{7}-e^{h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7}+\frac{2\pi }{7})}{1+e^{-2h\cos \frac{2\pi }{7}}-2e^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill T_5& =e^{-\cos {\displaystyle \frac{3\pi }{7}}}\cos (\sin {\displaystyle \frac{3\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}\cos \frac{3\pi }{7}-e^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7}+\frac{3\pi }{7})-e^{-2h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\cos (\sin \frac{3\pi }{7}-\frac{3\pi }{7})\\ +e^{-h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\cos (\sin \frac{3\pi }{7}+h\sin \frac{3\pi }{7}-\frac{3\pi }{7})\end{array}}{1+e^{-2h\cos \frac{3\pi }{7}}-2e^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7})}},\hfill \\ \hfill T_6& =1-2T{\displaystyle \frac{\cos \frac{4\pi }{7}-e^{h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7}+\frac{4\pi }{7})}{1+e^{-2h\cos \frac{4\pi }{7}}-2e^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill T_7& =e^{-\cos {\displaystyle \frac{5\pi }{7}}}\cos (\sin {\displaystyle \frac{5\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}\cos \frac{5\pi }{7}-e^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7}+\frac{5\pi }{7})-e^{-2h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\cos (\sin \frac{5\pi }{7}-\frac{5\pi }{7})\\ +e^{-h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\cos (\sin \frac{5\pi }{7}+h\sin \frac{5\pi }{7}-\frac{5\pi }{7})\end{array}}{1+e^{-2h\cos \frac{5\pi }{7}}-2e^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7})}},\hfill \\ \hfill T_8& =1-2T{\displaystyle \frac{\cos \frac{6\pi }{7}-e^{h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7}+\frac{6\pi }{7})}{1+e^{-2h\cos \frac{6\pi }{7}}-2e^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill T_9& =-e^{-\cos {\displaystyle \frac{\pi }{7}}}\sin (\sin {\displaystyle \frac{\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}\sin \frac{\pi }{7}-e^{-h\cos \frac{\pi }{7}}\sin (h\sin \frac{\pi }{7}+\frac{\pi }{7})+e^{-\cos \frac{\pi }{7}-2h\cos \frac{\pi }{7}}\sin (\sin \frac{\pi }{7}-\frac{\pi }{7})\\ -e^{-h\cos \frac{\pi }{7}-\cos \frac{\pi }{7}}\sin (\sin \frac{\pi }{7}+h\sin \frac{\pi }{7}-\frac{\pi }{7})\end{array}}{1+e^{-2h\cos \frac{\pi }{7}}-2e^{-h\cos \frac{\pi }{7}}\cos (h\sin \frac{\pi }{7})}},\hfill \\ \hfill T_{10}& =T{\displaystyle \frac{e^{-h\cos \frac{2\pi }{7}}\sin (h\sin \frac{2\pi }{7}+\frac{2\pi }{7})-\sin \frac{2\pi }{7}}{1+e^{-2h\cos \frac{2\pi }{7}}-2e^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill T_{11}& =-e^{-\cos {\displaystyle \frac{3\pi }{7}}}\sin (\sin {\displaystyle \frac{3\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}\sin \frac{3\pi }{7}-e^{-h\cos \frac{3\pi }{7}}\sin (h\sin \frac{3\pi }{7}+\frac{3\pi }{7})+e^{-\cos \frac{3\pi }{7}-2h\cos \frac{3\pi }{7}}\sin (\sin \frac{3\pi }{7}-\frac{3\pi }{7})\\ -e^{-h\cos \frac{3\pi }{7}-\cos \frac{3\pi }{7}}\sin (\sin \frac{3\pi }{7}+h\sin \frac{3\pi }{7}-\frac{3\pi }{7})\end{array}}{1+e^{-2h\cos \frac{3\pi }{7}}-2e^{-h\cos \frac{3\pi }{7}}\cos (h\sin \frac{3\pi }{7})}},\hfill \\ \hfill T_{12}& =T{\displaystyle \frac{e^{-h\cos \frac{4\pi }{7}}\sin (h\sin \frac{4\pi }{7}+\frac{4\pi }{7})-\sin \frac{4\pi }{7}}{1+e^{-2h\cos \frac{4\pi }{7}}-2e^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill T_{13}& =-e^{-\cos {\displaystyle \frac{5\pi }{7}}}\sin (\sin {\displaystyle \frac{5\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}\sin \frac{5\pi }{7}-e^{-h\cos \frac{5\pi }{7}}\sin (h\sin \frac{5\pi }{7}+\frac{5\pi }{7})+e^{-\cos \frac{5\pi }{7}-2h\cos \frac{5\pi }{7}}\sin (\sin \frac{5\pi }{7}-\frac{5\pi }{7})\\ -e^{-h\cos \frac{5\pi }{7}-\cos \frac{5\pi }{7}}\sin (\sin \frac{5\pi }{7}+h\sin \frac{5\pi }{7}-\frac{5\pi }{7})\end{array}}{1+e^{-2h\cos \frac{5\pi }{7}}-2e^{-h\cos \frac{5\pi }{7}}\cos (h\sin \frac{5\pi }{7})}},\hfill \\ \hfill T_{14}& =T{\displaystyle \frac{e^{-h\cos \frac{6\pi }{7}}\sin (h\sin \frac{6\pi }{7}+\frac{6\pi }{7})-\sin \frac{6\pi }{7}}{1+e^{-2h\cos \frac{6\pi }{7}}-2e^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill T_{15}& =1-{\displaystyle \frac{1}{e}}-T{\displaystyle \frac{e^{h+1}+1}{e^{h+1}+e}},\hfill \\ \hfill T_{16}& =\cos {\displaystyle \frac{2\pi }{7}}-e^{-\cos {\displaystyle \frac{2\pi }{7}}}\cos (\sin {\displaystyle \frac{2\pi }{7}}+{\displaystyle \frac{2\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}1-e^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})+e^{-2h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\cos (\sin \frac{2\pi }{7})\\ -e^{-h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\cos (\sin \frac{2\pi }{7}+h\sin \frac{2\pi }{7})\end{array}}{1+e^{-2h\cos \frac{2\pi }{7}}-2e^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill T_{17}& =\cos {\displaystyle \frac{4\pi }{7}}-e^{-\cos {\displaystyle \frac{4\pi }{7}}}\cos (\sin {\displaystyle \frac{4\pi }{7}}+{\displaystyle \frac{4\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}1-e^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})+e^{-2h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\cos (\sin \frac{4\pi }{7})\\ -e^{-h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\cos (\sin \frac{4\pi }{7}+h\sin \frac{4\pi }{7})\end{array}}{1+e^{-2h\cos \frac{4\pi }{7}}-2e^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill T_{18}& =\cos {\displaystyle \frac{6\pi }{7}}-e^{-\cos {\displaystyle \frac{6\pi }{7}}}\cos (\sin {\displaystyle \frac{6\pi }{7}}+{\displaystyle \frac{6\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}1-e^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})+e^{-2h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\cos (\sin \frac{6\pi }{7})\\ -e^{-h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\cos (\sin \frac{6\pi }{7}+h\sin \frac{6\pi }{7})\end{array}}{1+e^{-2h\cos \frac{6\pi }{7}}-2e^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}},\hfill \\ \hfill T_{19}& =\sin {\displaystyle \frac{2\pi }{7}}-e^{-\cos {\displaystyle \frac{2\pi }{7}}}\sin (\sin {\displaystyle \frac{2\pi }{7}}+{\displaystyle \frac{2\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}{e}^{-h\cos \frac{2\pi }{7}}\sin (h\sin \frac{2\pi }{7})+e^{-2h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\sin (\sin \frac{2\pi }{7})\\ -e^{-h\cos \frac{2\pi }{7}-\cos \frac{2\pi }{7}}\sin (\sin \frac{2\pi }{7}+h\sin \frac{2\pi }{7})\end{array}}{1+e^{-2h\cos \frac{2\pi }{7}}-2e^{-h\cos \frac{2\pi }{7}}\cos (h\sin \frac{2\pi }{7})}},\hfill \\ \hfill T_{20}& =\sin {\displaystyle \frac{4\pi }{7}}-e^{-\cos {\displaystyle \frac{4\pi }{7}}}\sin (\sin {\displaystyle \frac{4\pi }{7}}+{\displaystyle \frac{4\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}{e}^{-h\cos \frac{4\pi }{7}}\sin (h\sin \frac{4\pi }{7})+e^{-2h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\sin (\sin \frac{4\pi }{7})\\ -e^{-h\cos \frac{4\pi }{7}-\cos \frac{4\pi }{7}}\sin (\sin \frac{4\pi }{7}+h\sin \frac{4\pi }{7})\end{array}}{1+e^{-2h\cos \frac{4\pi }{7}}-2e^{-h\cos \frac{4\pi }{7}}\cos (h\sin \frac{4\pi }{7})}},\hfill \\ \hfill T_{21}& =\sin {\displaystyle \frac{6\pi }{7}}-e^{-\cos {\displaystyle \frac{6\pi }{7}}}\sin (\sin {\displaystyle \frac{6\pi }{7}}+{\displaystyle \frac{6\pi }{7}})-T{\displaystyle \frac{\begin{array}{c}{e}^{-h\cos \frac{6\pi }{7}}\sin (h\sin \frac{6\pi }{7})+e^{-2h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\sin (\sin \frac{6\pi }{7})\\ -e^{-h\cos \frac{6\pi }{7}-\cos \frac{6\pi }{7}}\sin (\sin \frac{6\pi }{7}+h\sin \frac{6\pi }{7})\end{array}}{1+e^{-2h\cos \frac{6\pi }{7}}-2e^{-h\cos \frac{6\pi }{7}}\cos (h\sin \frac{6\pi }{7})}}.\hfill \end{array}$$

Then, the second problem is solved. Finally, we obtain the optimal quadrature formula as the form (2) in the sense of Sard in the space $W_2^{(7,0)}$.