Here I am going to introduce some easy results on some criteria for interchanging the order of integration which are not covered by the classical Fubini theorem. Though both statements and proofs are weak and easy, it often reduces our burden to large extent. Let $f$ be a locally integrable function on $(0, \infty)$. That is, $f$ is a measurable function which is integrable on any compact subset..
Problem 1. Prove that[각주:1] \begin{equation*} \sum_{n=1}^{\infty} \frac{1}{n^4 \binom{2n}{n}} = \frac{17\pi^4}{3240}. \tag{1} \end{equation*} Proof. We divide the proof into several steps. 1. Reduction to an integral representation Let $S$ denote the summation in question. By using the Lemma 1 in Today's Calculation 29, we can represent $S$ as an integral. Then by the successive application of i..
Problem 1. Show that[각주:1] \begin{equation*} \int_{0}^{\infty} x \left\{ (2S(x) - 1)^2 + (2C(x) - 1)^2 \right\}^{2} \, \mathrm{d}x = \frac{16 \log 2 - 8}{\pi^2}, \tag{1} \end{equation*} where $S(x)$ and $C(x)$ denote the Fresnel integrals defined by \begin{align*} S(x) = \int_{0}^{x} \sin \left( \tfrac{1}{2} \pi t^2 \right) \, \mathrm{d}t \quad \text{and} \quad C(x) = \int_{0}^{x} \cos \left( \t..
Problem 1. Show that[각주:1] the product \begin{equation*} P = \left( \frac{2}{1} \right)^{\frac{1}{8}} \left( \frac{2 \cdot 2}{1 \cdot 3} \right)^{\frac{3}{16}} \left( \frac{2 \cdot 2 \cdot 2 \cdot 4}{1 \cdot 3 \cdot 3 \cdot 3} \right)^{\frac{6}{32}} \left( \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{1 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5} \right)^{\frac{10..
Residue 계산하면 금방 나오는 쉬운 계산입니다. Proposition 1. We have \begin{equation*} \int_{0}^{\infty} \frac{x^3}{(x^4 + 1)(e^{2\pi x} - 1)} \, \mathrm{d}x = \frac{\gamma}{2} + \frac{\pi}{4} \frac{\sin \pi\sqrt{2}}{\cosh \pi\sqrt{2} - \cos \pi\sqrt{2}} - \frac{1}{8} \sum_{\omega^{4} = -1} H_{\omega}, \end{equation*} where $H_s = \gamma + \psi_{0}(1+s)$ is the analytic extension of the harmonic number.
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