Math Problems Vault

Problem 5512

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\ctg (z-a)+\ctg a = \sum _{n=-\infty }^{\infty } \left(\frac{1}{z-a-n\pi }+\frac{1}{a+n\pi }\right).

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Problem 5513

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{1}{\sin z} = \frac{1}{z}+\sum _{n=1}^{\infty } \frac{(-1)^{n}2z}{z^{2}-n^{2}\pi ^{2}}.

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Problem 5514

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{1}{\cos z} = \pi \sum _{n=1}^{\infty } \frac{(-1)^{n}(2n-1)}{z^{2}-(2n-1)^{2}\frac{\pi ^{2}}{4}}.

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Problem 5515

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{\sin z}{\cos ^{2} z} = \sum _{n=-\infty }^{\infty } \frac{(-1)^{n}}{\left(z-(2n+1)\frac{\pi }{2}\right)^{2}}.

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Problem 5516

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{\pi ^{2}}{\cos ^{2} \pi z} = \sum _{n=-\infty }^{\infty } \frac{1}{\left(z+\frac{1}{2}-n\right)^{2}}.

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Problem 5517

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{e^{az}}{e^{z}-1} = \frac{1}{z}+\sum _{n=1}^{\infty } \frac{2z\cos 2\pi na-4\pi n\sin 2\pi na}{z^{2}+4n^{2}\pi ^{2}}; \quad 0 < a < 1.

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Problem 5518

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{\sin az}{\sin \pi z} = \frac{2}{\pi }\sum _{n=1}^{\infty } \frac{(-1)^{n}n\cdot \sin na}{z^{2}-n^{2}}; \quad -\pi < a < \pi .

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Problem 5519

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{\ch az}{\sh \pi z} = \frac{1}{\pi z}+\frac{1}{\pi }\sum _{n=1}^{\infty } \frac{(-1)^{n}2z\cos na}{z^{2}+n^{2}}; \quad -\pi \leqslant a \leqslant \pi .

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Problem 5520

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{1}{\ch 2z-\cos 2z} = \frac{1}{4z^{2}}+4z^{2}\sum _{n=1}^{\infty } \frac{(-1)^{n}n}{\sh \pi n}\cdot \frac{1}{4z^{4}+n^{4}\pi ^{4}}.

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Problem 5521

Prove the validity of the partial-fraction expansion, which holds at all points where the function being expanded is regular:

\frac{z\sin z}{\sin z-z\cos z} = \frac{3}{z}+\sum _{n=1}^{\infty } \frac{2z}{z^{2}-\lambda _{n}^{2}}; \quad \tg \lambda _{n} = \lambda _{n} > 0.

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Problem 5522

Prove the validity of the infinite-product expansion:

\cos \frac{\pi z}{2} = \prod _{n=0}^{\infty } \left(1-\frac{z^{2}}{(2n+1)^{2}}\right).

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Problem 5523

Prove the validity of the infinite-product expansion:

\frac{\sin z-a}{\sin a} = e^{-z\ctg a}\prod _{n=-\infty }^{\infty } \left(1-\frac{z}{a+n\pi }\right)\cdot e^{\frac{z}{a+n\pi }}.

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Problem 5524

Prove the validity of the infinite-product expansion:

\cos \pi z-\cos \pi a = -\frac{\pi ^{2}}{2}(z^{2}-a^{2})\prod _{n=1}^{\infty } \left(1-\left(\frac{z+a}{2n}\right)^{2}\right)\cdot \left(1-\left(\frac{z-a}{2n}\right)^{2}\right).

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Problem 5525

Prove the validity of the infinite-product expansion:

e^{z}-e^{a} = e^{\frac{z+a}{2}}\cdot (z-a)\cdot \prod _{n=1}^{\infty } \left(1+\left(\frac{z-a}{2\pi n}\right)^{2}\right).

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Problem 5526

Prove the validity of the infinite-product expansion:

\ch z-\cos z = z^{2}\prod _{n=1}^{\infty } \left(1+\frac{z^{4}}{4\pi ^{4}n^{4}}\right).

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Problem 5527

Prove the validity of the infinite-product expansion:

\sin z-z\cos z = \frac{z^{2}}{3}\prod _{n=1}^{\infty } \left(1-\frac{z^{2}}{\lambda _{n}^{2}}\right); \quad \tg \lambda _{n} = \lambda _{n} > 0.

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Problem 5528

Prove the validity of the infinite-product expansion:

\frac{\sin \pi z}{\pi z(1-z)} = \prod _{n=1}^{\infty } \left(1+\frac{z-z^{2}}{n+n^{2}}\right).

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Problem 5529

Prove the validity of the infinite-product expansion:

\prod _{n=1}^{\infty } \left(1-\frac{z}{a^{n}}\right) = \sum _{n=0}^{\infty } c_{n}z^{n}; \quad \left|a\right| > 1; \quad \prod _{\nu =1}^{n} (a^{\nu }-1) = \frac{(-1)^{n}}{c_{n}}.

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Problem 5530

Prove the validity of the infinite-product expansion:

\sum _{\nu =0}^{\infty } \frac{z^{\nu }}{(3\nu )!} = \prod _{\nu =1}^{\infty } \left(1+\frac{z}{8r_{\nu }^{3}}\right).

2\cos r_{\nu }\sqrt{3}+e^{-3r_{\nu }} = 0, \quad r_{\nu } > 0.

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Problem 5531

For $\mathrm{Re}\left(s\right) > 0$ , Euler's gamma function has the integral representation $\Gamma (s) = \int_{0}^{\infty } e^{-x}x^{s-1}\, dx$ . It is easy to prove that for $0 < x < n$ the formulas

e^{-x}\left(1-\frac{x^{2}}{n}\right) < \left(1-\frac{x}{n}\right)^{n} < e^{-x}

hold. Using this, prove Euler's representation of $\Gamma (s)$ as an infinite product:

\Gamma (s) = \lim _{n\to \infty } \frac{1\cdot 2\cdot 3\cdots n\cdot n^{s}}{s(s+1)\cdots (s+n)}.

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Problem 5532

Using the preceding representation $\Gamma (s) = \lim_{n\to \infty } \dfrac {1\cdot 2\cdot 3\cdots n\cdot n^{s}}{s(s+1)\cdots (s+n)}$ (problem 5531), as well as the formula

\lim _{n\to \infty } \left(1+\frac{1}{2}+\frac{1}{3}+\ldots +\frac{1}{n}-\ln n\right) = C,

where $C$ is Euler's constant, prove the Weierstrass formula

\frac{1}{\Gamma (s)} = se^{Cs}\prod _{\nu =1}^{\infty } \left(1+\frac{s}{\nu }\right)e^{-\frac{s}{\nu }}.

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Problem 5533

Prove the identity:

\prod _{n=1}^{\infty } \frac{n(a+b+n)}{(a+n)(b+n)} = \frac{\Gamma (a+1)\Gamma (b+1)}{\Gamma (a+b+1)}.

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Problem 5534

Prove the identity:

\frac{\Gamma '(s)}{\Gamma (s)} = -C-\frac{1}{s}+\sum _{n=1}^{\infty } \left(\frac{1}{n}-\frac{1}{n+s}\right).

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Problem 5535

Prove the identity:

\left(\frac{\Gamma '(s)}{\Gamma (s)}\right)' = \sum _{\nu =0}^{\infty } \frac{1}{(s+\nu )^{2}}.

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Problem 5536

Prove the identity:

\ln s\Gamma (s) = \left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-C\right)s+\ln \frac{1\cdot 2\cdot 3\cdot 4}{(s+1)(s+2)(s+3)(s+4)}+\sum _{\nu =2}^{\infty } \frac{(-1)^{\nu }s^{\nu }}{\nu }\sigma _{\nu },

where $\sigma_{\nu } = \dfrac {1}{5^{\nu }}+\dfrac {1}{6^{\nu }}+\dfrac {1}{7^{\nu }}+\ldots$ .

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Problem 5537

Prove the identity:

\lg _{10} x\Gamma (x) = \lg _{10} \frac{1\cdot 2\cdot 3\cdot 4}{(1+x)(2+x)(3+x)(4+x)}+0{,}65409856x-0{,}00005089x^{5}+

+0{,}04805967x^{2}+0{,}00000740x^{6}-0{,}00353152x^{3}-0{,}00000116x^{7}+

+0{,}00038775x^{4}+0{,}00000019x^{8}-0{,}00000003x^{9}\ldots

Here $\left|x\right| < 5$ .

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Problem 5538

From Euler's product for $\Gamma (s)$ it follows that

S = -\sum _{\nu =0}^{m} \ln (s+\nu )+\ln m!+s\ln m \to \ln \Gamma (s).

If the factor $-1$ occurring in the sum is replaced using the formula $-1 = \left(s+\nu -\frac{1}{2}\right)-\left(s+\nu +\frac{1}{2}\right)$ , one easily obtains the identities:

S = \sum _{\nu =0}^{m} \left(\left(s+\nu +\frac{1}{2}\right)\ln \frac{s+\nu +1}{s+\nu }-1\right)+\left(s-\frac{1}{2}\right)\ln s-\left(s+m+\frac{1}{2}\right)\ln (s+m+1)+

+\ln m!+s\ln m+m+1 =

= \sum _{\nu =0}^{m} \left(\left(s+\nu +\frac{1}{2}\right)\ln \frac{s+\nu +1}{s+\nu }-1\right)+\left(s-\frac{1}{2}\right)\ln s-s+m+\ln m!-m\ln m-\frac{1}{2}\ln m+\frac{M}{m},

where $M$ is bounded as $m \to \infty$ . Using this, obtain Gudermann's series:

\ln \Gamma (s) = \left(s-\frac{1}{2}\right)\ln s-s+A+\sum _{\nu =0}^{\infty } \left(\left(s+\nu +\frac{1}{2}\right)\ln \frac{s+\nu +1}{s+\nu }-1\right),

where $A$ is a corresponding constant, and the identity holds for all $s$ other than the numbers $0, -1, -2, -3, \ldots$ .

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Problem 5539

Applying the expansion of $\cos \dfrac {\pi z}{2}$ as an infinite product (problem 5522), prove the Wallis formula:

\frac{\pi }{2} = \lim _{n\to \infty } \left(\frac{2\cdot 4\cdot 6\cdots 2n}{3\cdot 5\cdot 7\cdots (2n+1)}\right)^{2}\cdot (2n+2),

and, using it, prove that the constant $A$ in Gudermann's series (problem 5538) equals $\ln \sqrt{2\pi }$ .

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Problem 5540

Using Gudermann's series (problem 5538), prove that

\ln \Gamma (s)-\left(\left(s-\frac{1}{2}\right)\ln s-s\right) \to \ln \sqrt{2\pi },

if the distance of the point $s$ from the negative part of the real axis grows without bound, while $\mathrm{Re}\left(s\right)$ is bounded below.

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Problem 5541

Prove the validity of the expansion:

\frac{\Gamma (a)\Gamma (x)}{\Gamma (a+x)} = \frac{1}{x}+\sum _{n=1}^{\infty } \frac{(-1)^{n}(a-1)(a-2)\ldots (a-n)}{1\cdot 2\cdot 3\cdots n}\cdot \frac{1}{x+n}; \quad a > 0.

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