Golden Ratio and a Ramanujan-Type Integral
Abstract
:1. Introduction
2. Proof of Equation (3)
- Step I: Rewriting the sum side of Equation (7)Our goal is to show that the left-hand side of Equation (7) is the same asIndeed, let us consider the sum involving x in Equation (8). We break it up according to , and :
- Step II: Identifying the poles of the infinite product in Equation (7)Let us denote by the right-hand side of Equation (7). First we treat as a function of x. At the same time, we will treat z as a parameter that is not an integral power of q. We claim that has poles of order two atIndeed, the denominator in implies is a pole of order 2. Similarly, the denominator implies is a pole of order 2. This proves our claim.Next we want to find the partial fraction expansion of . To this end, we need to determine the symmetries of .
- Step III: Exploring the symmetries ofReaders can easily verify the following:
- Step IV: Finding the partial fraction expansion ofOur goal is to prove thatHere is a “remainder” term that has a Laurent expansion in x. Note that, by comparing with Step I above, Equation (12) implies that we have “half” of Equation (7).First we show that . The part of that contributes to comes solely from the overall prefactor (note that the infinite product becomes 1 as ). Regarding this prefactor, we note thatThis implies the principal part at isFor , we have
- Step V: DeterminingThis is the final step: let us show thatWe recall that both H and G were defined in Equation (12). Previously, represented what cannot be determined by understanding the pole structure of . What Equation (15) says is that, can indeed be written as something known (i.e., G)—but there is a catch: the argument of G on the right-hand side of Equation (15) is z, not x. In fact, the same equation tells us the is independent of x. Let us turn to the proof.Since is a Laurent expansion in x (cf. the sentence right after Equation (12)), we can write it asFirst we note that satisfiesIndeed,This, with Equation (12), impliesBy Equations (9) and (18), we have
I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.
3. Final Remarks
Acknowledgements
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Chan, H.-C. Golden Ratio and a Ramanujan-Type Integral. Axioms 2013, 2, 58-66. https://doi.org/10.3390/axioms2010058
Chan H-C. Golden Ratio and a Ramanujan-Type Integral. Axioms. 2013; 2(1):58-66. https://doi.org/10.3390/axioms2010058
Chicago/Turabian StyleChan, Hei-Chi. 2013. "Golden Ratio and a Ramanujan-Type Integral" Axioms 2, no. 1: 58-66. https://doi.org/10.3390/axioms2010058
APA StyleChan, H. -C. (2013). Golden Ratio and a Ramanujan-Type Integral. Axioms, 2(1), 58-66. https://doi.org/10.3390/axioms2010058