Proving identities involving the gamma and beta functions

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Problem 18.31 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.3 "Convolution of generalized functions"

Compute the convolutions:

(a)

$\delta_{0} * \varphi , \varphi \in \mathcal{D}$ ;

(b)

$\delta_{0}^{(n)} * \varphi , \varphi \in \mathcal{D}$ ;

(c)

$\delta_{0}^{(n)} * \delta_{0}^{(k)}$ ;

(d)

$\delta_{0}^{(n)} * x_{+}^{k}(x)$ ;

(e)

$\left(\theta (x) e^{a x}\right) *\left(\theta (x) e^{b x}\right)$ ;

(f)

$x_{+}^{k} * x_{+}^{n}$ ;

(g)

$x_{+}^{\alpha } * x_{+}^{\beta }, \operatorname {Re} \alpha > -1, \operatorname {Re} \beta > -1$ .

Problem 3872 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Prove the identity

\int _{0}^{1}\dfrac {dx}{\sqrt{1-x^{4}}} \cdot \int _{0}^{1}\dfrac {x^{2}\, dx}{\sqrt{1-x^{4}}}=\dfrac {\pi }{4}.

Problem 3873 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Prove the identity

\int _{0}^{+\infty }e^{-x^{4}}\, dx \cdot \int _{0}^{+\infty }x^{2}e^{-x^{4}}\, dx=\dfrac {\pi }{8\sqrt{2}}.

Problem 3874 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Prove the identity

\prod _{m=1}^{n}\int _{0}^{+\infty }x^{m-1}e^{-x^{n}}\, dx=\left(\dfrac {1}{n}\right)^{n+\frac{1}{2}}(2\pi )^{\frac{n-1}{2}}.

Problem 3875 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Prove the identity

\lim _{n \to \infty }\int _{0}^{+\infty }e^{-x^{n}}\, dx=1.

Problem 3876 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Using the identity $\dfrac {1}{x^{m}}=\dfrac {1}{\Gamma (m)}\displaystyle \int_{0}^{+\infty }t^{m-1}e^{-xt}\, dt$ ( $x>0$ ), compute the integral

\int _{0}^{+\infty }\dfrac {\cos ax}{x^{m}}\, dx \quad (0<m<1).

Problem 3877 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

Using the identity $\dfrac {1}{x^{m}}=\dfrac {1}{\Gamma (m)}\displaystyle \int_{0}^{+\infty }t^{m-1}e^{-xt}\, dt$ ( $x>0$ ), compute the integral

\int _{0}^{+\infty }\dfrac {\sin ax}{x^{m}}\, dx \quad (0<m<2).

Problem 3878 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.4 "Euler Integrals"

(a)(b)

Prove the Euler formulas:

(a)

$\displaystyle \int_{0}^{+\infty }t^{x-1}e^{-\lambda t\cos \alpha }\cos (\lambda t\sin \alpha )\, dt=\dfrac {\Gamma (x)}{\lambda^{x}}\cos \alpha x$ ;

(b)

$\displaystyle \int_{0}^{+\infty }t^{x-1}e^{-\lambda t\cos \alpha }\sin (\lambda t\sin \alpha )\, dt=\dfrac {\Gamma (x)}{\lambda^{x}}\sin \alpha x$ ,

where $\lambda >0$ , $x>0$ , $-\dfrac {\pi }{2}<\alpha <\dfrac {\pi }{2}$ .

Problem 4720 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Another Euler integral, the beta function $B(p,q)$ , is expressed in terms of the gamma function. By definition, for $p > 0$ and $q > 0$ we have:

B(p,q) = \int _{0}^{1} x^{p-1}(1-x)^{q-1}\, dx .

There is a formula

B(p,q) = \frac{\Gamma (p)\Gamma (q)}{\Gamma (p+q)} .

By a change of variables, integrals of the form

\int _{0}^{\infty } \frac{x^{m}}{(a+bx^{n})^{p}}\, dx \quad \text{for } a > 0 \text{ and } b > 0

reduce to the beta function. It is often useful to replace the fraction $\dfrac {1}{a^{n}}$ , where $n > 0$ , $a > 0$ , by means of the formula

\frac{1}{a^{n}} = \frac{1}{\Gamma (n)} \int _{0}^{\infty } e^{-ax} x^{n-1}\, dx .

Prove the equalities, for positive integers $n$ and $m$ :

\int _{0}^{\infty } \frac{\sin ^{2m} x}{x^{p}}\, dx = \frac{(2m)!}{\Gamma (p)} \int _{0}^{\infty } \frac{t^{p-2}\, dt}{(t^{2}+2^{2})(t^{2}+4^{2})\ldots (t^{2}+4m^{2})},

1 < p < 2m+1 .

Problem 4754 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Prove the equalities:

\int _{0}^{\infty } e^{-x^{4}}\, dx \int _{0}^{\infty } x^{2} e^{-x^{4}}\, dx = \frac{\pi }{8\sqrt{2}} .

Problem 4755 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Prove the equalities:

\int _{0}^{1} \frac{dx}{\sqrt{1-x^{4}}} \int _{0}^{1} \frac{x^{2}\, dx}{\sqrt{1-x^{4}}} = \frac{\pi }{4} .

Problem 4756 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Prove the equalities:

\int _{0}^{\infty } x^{p-1} \cos ax\, dx = \frac{1}{a^{p}}\Gamma (p)\cos \frac{\pi p}{2}; \quad 0 < p < 1 .

Problem 4757 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Prove the equalities:

\int _{0}^{\infty } x^{p-1} \sin ax\, dx = \frac{1}{a^{p}}\Gamma (p)\sin \frac{\pi p}{2}; \quad -1 < p < 1 .

Problem 4758 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.7 "Euler Integrals"

Prove the equalities:

\int _{0}^{\infty } \frac{x^{p-\frac{1}{2}}\, dx}{(x^{2}+ax+b)^{p}} = \frac{\sqrt{\pi }}{2\sqrt{b}\left(a+2\sqrt{b}\right)^{p-\frac{3}{2}}} \cdot \frac{\Gamma \left(p-\frac{1}{2}\right)}{\Gamma (p)}; \quad b > 0,

a+2\sqrt{b} > 0, \quad p > \frac{1}{2} .

Problem 4773 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.8 "Miscellaneous Problems"

Prove that for $f(x) = e^{-\sqrt[4]{x}} \sin x \sqrt[4]{x}$ , for all integers $n \geqslant 0$ , the equality $\int_{0}^{\infty } f(x) x^{n}\, dx = 0$ holds.

Problem 3.16.20 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.16.2 "Problems"

(1)(2)

Prove Euler's formulas:

(1)

$\int_{0}^{+\infty } t^{x-1} e^{-\lambda t \cos a} \cos (\lambda t \sin \alpha ) d t = \frac{\Gamma (x)}{\lambda^{x}} \cos \alpha x$

(2)

$\int_{0}^{+\infty } t^{x-1} e^{-\lambda t \cos \alpha } \sin (\lambda t \sin \alpha ) d t = \frac{\Gamma (x)}{\lambda^{x}} \sin \alpha x, \quad \lambda > 0, \quad x > 0$ , $\left|\alpha \right| < \pi / 2$ .

Problem VI.2.6 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.2 "Laplace's Method"

(a)(b)(c)(d)

We assume that $\varphi$ is a positive nonincreasing function on the interval $[a, b)(-\infty <a<b \leqslant +\infty )$ , with $\varphi (a)=\lim_{x \rightarrow a+0} \varphi (x)=1$ a strict maximum of $\varphi$ on $[a, b)$ . We also assume that $f \in C((a, b)), f \geqslant 0$ on $(a, b)$ and $f \not\equiv 0$ near the point $a$ , and that the integral $\int_{a}^{b} f(x) \varphi^{A}(x) d x$ is finite for sufficiently large $A$ . Let $\Phi (A)=\int_{a}^{b} \varphi^{A}(x) d x$ . Prove that

(a)

if for some $L, p>0$ the inequality $\varphi (x) \geqslant 1-L(x-a)^{p}$ holds near the point $a$ , then

A^{1 / p} \Phi (A) \geqslant \frac{\Gamma \left(1+\frac{1}{p}\right)}{L^{1 / p}}+o(1) \quad \text{ as } \quad A \rightarrow +\infty ;

(b)

if $\frac{1-\varphi (x)}{(x-a)^{p}} \rightarrow 0$ as $x \rightarrow a+0$ , then $A^{1 / p} \Phi (A) \rightarrow +\infty$ as $A \rightarrow +\infty$ ;

(c)

if for some $l, p>0$ the inequality $\varphi (x) \leqslant 1-l(x-a)^{p}$ holds near the point $a$ , then

A^{1 / p} \Phi (A) \leqslant \frac{\Gamma \left(1+\frac{1}{p}\right)}{l^{1 / p}}+o(1) \quad \text{ as } \quad A \rightarrow +\infty ;

(d)

if $\frac{1-\varphi (x)}{(x-a)^{p}} \rightarrow +\infty$ as $x \rightarrow a+0$ , then

A^{1 / p} \Phi (A) \rightarrow 0 \quad \text{ as } \quad A \rightarrow +\infty .

Problem VI.2.7 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.2 "Laplace's Method"

We assume that $\varphi$ is a positive nonincreasing function on the interval $[a, b)(-\infty <a<b \leqslant +\infty )$ , with $\varphi (a)=\lim_{x \rightarrow a+0} \varphi (x)=1$ a strict maximum of $\varphi$ on $[a, b)$ . We also assume that $f \in C((a, b)), f \geqslant 0$ on $(a, b)$ and $f \not\equiv 0$ near the point $a$ , and that the integral $\int_{a}^{b} f(x) \varphi^{A}(x) d x$ is finite for sufficiently large $A$ . Let $1-\varphi (x) \underset {x \rightarrow a+0}{\sim } C(x-a)^{p}(C, p>0)$ . Prove the Laplace asymptotic formula

\Phi (A)=\int _{a}^{b} \varphi ^{A}(x) d x \underset {A \rightarrow +\infty }{\sim } \frac{\Gamma \left(1+\frac{1}{p}\right)}{(A C)^{1 / p}}

In particular,

\Phi (A) \sim \frac{1}{A C} \quad \text{ for } p=1, \quad \Phi (A) \sim \frac{1}{2} \sqrt{\frac{\pi }{A C}} \quad \text{ for } p=2 .

Supplement the Laplace formula by proving that for any $q>-1$

\int _{a}^{b}(x-a)^{q} \varphi ^{A}(x) d x \underset {A \rightarrow +\infty }{\sim } \frac{1}{q+1} \frac{\Gamma \left(1+\frac{q+1}{p}\right)}{(A C)^{\frac{q+1}{p}}}

Problem VI.3.30 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.3 "Asymptotics of Sums"

(a)(b)

Find the asymptotics as $t \rightarrow 1-0$ of the sums of the series

(a)

$\sum n^{p} t^{n}(p>-1)$ ;

(b)

$\sum_{n \geqslant 2}(\ln n)^{p} t^{n}(p \in \mathbb {R})$ .

Problem 2.35 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 4 "Improper Integrals Between Infinite Limits"

We have

\lim _{t \to 1-0} \sqrt{1-t} \, (1 + t + t^{4} + t^{9} + \cdots + t^{n^{2}} + \cdots ) = \frac{\sqrt{\pi }}{2}

and more generally for $\alpha > 0$

\lim _{t \to 1-0} \sqrt[\alpha ]{1-t} \, (1 + t^{1^{\alpha }} + t^{2^{\alpha }} + t^{3^{\alpha }} + \cdots + t^{n^{\alpha }} + \cdots ) = \frac{1}{\alpha } \Gamma \left(\frac{1}{\alpha }\right) .

Problem 2.65 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7 "Multiple Integrals"

Let $\alpha_{1}, \alpha_{2}, \ldots , \alpha_{p}$ be arbitrary positive numbers, define

f_{v}(z) = 1^{\alpha _{v}-1} z + 2^{\alpha _{v}-1} z^{2} + \cdots + n^{\alpha _{v}-1} z^{n} + \cdots , \qquad v = 1, 2, \ldots , p

and

f_{1}(z) f_{2}(z) \cdots f_{p}(z) = \sum _{n=1}^{\infty } a_{n} z^{n} .

Show that

\lim _{n \to \infty } \frac{a_{n}}{n^{\alpha _{1}+\alpha _{2}+\cdots +\alpha _{p}-1}} = \idotsint x_{1}^{\alpha _{1}-1} x_{2}^{\alpha _{2}-1} \cdots x_{p-1}^{\alpha _{p-1}-1} (1 - x_{1} - x_{2} - \cdots - x_{p-1})^{\alpha _{p}-1} \, dx_{1} \, dx_{2} \cdots dx_{p-1},

where the integral is taken over the domain described by the $p$ inequalities $x_{1} \geq 0$ , $x_{2} \geq 0$ , , $x_{p-1} \geq 0$ , $x_{1} + x_{2} + \cdots + x_{p-1} \leq 1$ ( $p-1$ dimensional simplex).

Problem 3.151 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Cauchy's Formula"

We have for $0<\mathrm{Re}\left(s\right)<1$

\int _{0}^{\infty } x^{s-1}e^{-ix}\, dx = \Gamma (s)e^{-\frac{i\pi s}{2}}.

Problem 3.152 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Cauchy's Formula"

The equation

\int _{0}^{\infty } \frac{\sin (x^{n})}{x^{n}}\, dx = \frac{1}{n-1}\Gamma \left(\frac{1}{n}\right)\sin \left(\frac{n-1}{n}\frac{\pi }{2}\right)

holds for $n>1$ .

Problem 3.153 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Cauchy's Formula"

We find, assuming $\mu >0$ , $0<\alpha <\frac{\pi }{2}$ , $n=0,1,2,\ldots$ , that

\int _{0}^{\infty } e^{-x^{\mu }\cos \alpha }\sin (x^{\mu }\sin \alpha )x^{n}\, dx = \frac{1}{\mu }\Gamma \left(\frac{n+1}{\mu }\right)\sin \frac{(n+1)\alpha }{\mu }.

Note the special case $\alpha =\mu \pi$ .

Problem 3.154 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Cauchy's Formula"

Assume $\mu >0$ , $x>0$ , $\mu$ fixed, $x$ variable. Show that

\lim _{x\to \infty } x^{\mu +1}\int _{0}^{+\infty } e^{-t^{\mu }}\cos xt\, dt = \Gamma (\mu +1)\sin \frac{\mu \pi }{2}.

Problem 5.184 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 2 "Precise Determination of the Number of Zeros by Descartes' Rule of Signs"

Prove the formula, valid for $\omega >0$ , $-1+(m-1)\omega <\mathrm{Re}\left(z\right)<0$ ,

\frac{Q(z,\omega )}{\omega ^m} = \frac{\Gamma \left(\dfrac {1}{\omega }+1\right)}{\Gamma \left(\dfrac {1+z}{\omega }-m+1\right)\Gamma \left(-\dfrac {z}{\omega }\right)} \int _0^1 (1-t)^{\frac{1+z}{\omega }}t^{-\frac{z+\omega }{\omega }}f\left(\frac{t}{t-1}\right)\, dt

and find analogous formulae for $P(z,\omega )$ and $R(z,\omega )$ .

Problem 3.8.6 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.8 "Applications to Differential Equations"

For $0 < \lambda < n$ and $x \in \mathbb {R}^{n}$ , define

K_{\lambda }(x) = \left|x\right|^{-\lambda }

Show that

\hat{K}_{\lambda }(y) = c(n, \lambda ) K_{n-\lambda }(y) \quad \left(y \in \mathbb {R}^{n}\right) \tag {a}

where

c(n, \lambda ) = \frac{2^{n / 2-\lambda } \Gamma [(n-\lambda ) / 2]}{\Gamma (\lambda / 2)}

Suggestion: If $n < 2 \lambda < 2 n, K_{\lambda }$ is the sum of an $L^{1}$ -function and an $L^{2}$ -function. For these $\lambda$ , Equation (a) can be deduced from the homogeneity condition

K_{\lambda }(t x) = t^{-\lambda } K_{\lambda }(x) \quad \left(x \in \mathbb {R}^{n}, t > 0\right)

The case $0 < 2 \lambda < n$ follows from the inversion theorem (for tempered distributions). A passage to the limit gives the case $2 \lambda = n$ . The constants $c(n, \lambda )$ can be computed from $\int f \hat{\phi } = \int \hat{\delta } \phi$ , with $\phi (x) = \exp \left(-\left|x\right|^{2} / 2\right)$ .

Problem 24.2 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Prove the formulas:

(1)

$\int_{0}^{+\infty } x^{z} e^{-x^{2}} d x = \frac{1}{2} \Gamma \left(\frac{z+1}{2}\right) \quad (\mathrm{Re}\left(z\right) > -1) ;$

(2)

$\int_{0}^{+\infty } x^{z} e^{-x^{p}} d x = \frac{1}{p} \Gamma \left(\frac{z+1}{p}\right)(p > 0, \mathrm{Re}\left(z\right) > -1)$ ;

(3)

$\quad \int_{0}^{+\infty } t^{z-1} e^{-\zeta t} d t = \zeta^{-z} \Gamma (z) \quad \left(\mathrm{Re}\left(z\right) > 0,\left|\arg \zeta \right| < \frac{\pi }{2}\right)$ ;

(4)

$\quad \int_{0}^{+\infty } t^{z-1} \cos t d t = \Gamma (z) \cos \frac{\pi z}{2} \quad (0 < \mathrm{Re}\left(z\right) > 1)$ ;

(5)

$\quad \int_{0}^{+\infty } t^{z-1} \sin t d t =$

\begin{cases} \Gamma (z) \sin \frac{\pi z}{2}, & z \neq 0(-1 < \mathrm{Re}\left(z\right) < 1), \\ \frac{\pi }{2}, & z = 0 ; \end{cases}

(6)

$\int_{0}^{+\infty } \frac{\sin x^{p}}{x^{p}} d x = \frac{1}{p-1} \Gamma \left(\frac{1}{p}\right) \cos \frac{\pi }{2 p} \quad \left(p > \frac{1}{2}\right)$ ;

(7)

$\int_{0}^{+\infty } \frac{1-\cos x^{p}}{x^{2 p}} d x = \frac{p}{(p-1)(2 p-1)} \Gamma \left(\frac{1}{p}\right) \cos \frac{\pi }{2 p} \quad \left(p > \frac{1}{2}\right)$ ;

(8)

$\int_{0}^{+\infty } e^{-x^{p} \cos p \lambda } \cos \left(x^{p} \sin p \lambda \right) d x = \frac{1}{p} \Gamma \left(\frac{1}{p}\right) \cos \lambda \left(p > 0,-\frac{\pi }{2 p} < \lambda < \frac{\pi }{2 p}\right) ;$

(9)

$\int_{0}^{+\infty } x^{z-1} e^{-x^{p} \cos \lambda } \sin \left(x^{p} \sin \lambda \right) d x = \frac{1}{p} \Gamma \left(\frac{z}{p}\right) \sin \frac{\lambda z}{p}$ $\left(\mathrm{Re}\left(z\right) > 0, p > 1,-\frac{\pi }{2} < \lambda < \frac{\pi }{2}\right)$ .

Problem 24.3 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Prove the formulas:

(1)

$\int_{0}^{1} t^{\alpha -1}\left(1-t^{2}\right)^{\beta -1} d t = \frac{1}{2} \mathrm{~ B}\left(\frac{\alpha }{2}, \beta \right) \quad (\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0)$ ;

(2)

$\int_{0}^{1} t^{\alpha -1}\left(1-t^{p}\right)^{\beta -1} d t = \frac{1}{p} \mathrm{~ B}\left(\frac{\alpha }{p}, \beta \right)(p > 0, \mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0)$ ;

(3)

$\int_{0}^{+\infty } \frac{t^{\alpha -1} d t}{(1+t)^{\alpha +\beta }} = \mathrm{B}(\alpha , \beta ) \quad (\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0)$ ;

(4)

$\int_{0}^{+\infty } \frac{t^{\alpha -1} d t}{\left(1+t^{2}\right)^{\beta }} = \frac{1}{2} \mathrm{~ B}\left(\frac{\alpha }{2}, \beta -\frac{\alpha }{2}\right) \quad \left(\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\left(\beta -\frac{\alpha }{2}\right)\right) > 0\right)$ ;

(5)

$\int_{0}^{+\infty } \frac{t^{\alpha -1} d t}{\left(1+t^{p}\right)^{\beta }} = \frac{1}{p} \mathrm{~ B}\left(\frac{\alpha }{p}, \beta -\frac{\alpha }{p}\right)\left(p > 0, \mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\left(\beta -\frac{\alpha }{p}\right)\right) > 0\right)$ .

Problem 24.4 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

(1)(2)(3)

Prove the formulas:

(1)

$\int_{0}^{1} \frac{t^{\alpha -1}\left(1-t^{2}\right)^{\beta -1}}{\left(1+t^{2}\right)^{(\alpha / 2)+\beta }} d t = 2^{-(\alpha / 2)-1} \mathrm{~ B}\left(\frac{\alpha }{2}, \beta \right) \quad (\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0)$ ;

(2)

$\int_{0}^{1} \frac{t^{\alpha -1}\left(1-t^{p}\right)^{\beta -1}}{\left(1+t^{p}\right)^{(\alpha / p)+\beta }} d t = \frac{1}{p} 2^{-(\alpha / p)} \mathrm{B}\left(\frac{\alpha }{p}, \beta \right)$ (here $p > 0, \mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0$ );

(3)

$\int_{-1}^{1} \frac{(1+t)^{\alpha -1}(1-t)^{\beta -1}}{(t+a)^{\alpha +\beta }} d t = 2^{\alpha +\beta -1}(a+1)^{-\alpha }(a-1)^{-\beta } \mathrm{B}(\alpha , \beta )$ (here $a > 1, \mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0$ );

Problem 24.5 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

(1)

Let $\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0$ . Prove that

\int _{-1}^{1} \frac{(1+x)^{2 \alpha -1}(1-x)^{2 \beta -1}}{\left(1+x^{2}\right)^{\alpha +\beta }} d x = 2^{\alpha +\beta -2} \mathrm{~ B}(\alpha , \beta )

(2)

For $a > 0, \mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0$ , prove that

\int _{0}^{\frac{\pi }{2}} \frac{(\sin \varphi )^{2 \alpha -1}(\cos \varphi )^{2 \beta -1}}{\left(a+\sin ^{2} \varphi \right)^{\alpha +\beta }} d \varphi = \frac{1}{2}(a+1)^{-\alpha } a^{-\beta } B(\alpha , \beta ).

(3)

For $\left|\mathrm{Re}\left(\alpha \right)\right| < \left|\mathrm{Re}\left(\beta \right)\right|$ , prove that $\int_{0}^{\infty } e^{\alpha x}(\operatorname {ch} x)^{-\beta } d x = 2^{\beta -1} \mathrm{~ B}\left(\frac{\beta +\alpha }{2}, \frac{\beta -\alpha }{2}\right)$ .

(4)

For $\left|\mathrm{Re}\left(\alpha \right)\right| < \left|\mathrm{Re}\left(\beta \right)\right|$ , prove that

\int _{0}^{+\infty } \frac{\operatorname {ch} \alpha x}{(\operatorname {ch} x)^{\beta }} d x = 2^{\beta -1} \mathrm{~ B}\left(\frac{\beta +\alpha }{2}, \frac{\beta -\alpha }{2}\right).

(5)

For $\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\alpha +\beta \right) < 0$ , prove that

\int _{0}^{+\infty }(\operatorname {sh} x)^{2 \alpha -1}(\operatorname {ch} x)^{2 \beta -1} d x = \frac{1}{2} \mathrm{~ B}(\alpha ,-\alpha -\beta )

Problem 24.8 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

(1)(2)(3)(4)

Prove that

(1)

for any $\alpha$ and for $\mathrm{Re}\left(\beta \right) > 0$

\int _{0}^{\frac{\pi }{2}}(\cos \varphi )^{\beta -1} \cos \alpha \varphi d \varphi = 2^{-\beta } \sin \frac{\pi (\beta -\alpha )}{2} \mathrm{~ B}\left(\frac{\alpha -\beta +1}{2}, \beta \right) ;

(2)

for any $\alpha$ and for $\mathrm{Re}\left(\beta \right) > -\frac{1}{2}$

\int _{0}^{\pi / 2}(\cos \varphi )^{2 \beta } \cos ^{2} \alpha \varphi d \varphi = 2^{-2 \beta -2} \cos \pi (\alpha -\beta ) \mathrm{B}(\alpha -\beta , 2 \beta )+\frac{1}{2} \mathrm{~ B}\left(\beta +\frac{1}{2}, \frac{1}{2}\right) ;

(3)

for $\mathrm{Re}\left(\beta \right) < 1$ and $\mathrm{Re}\left(\alpha -\beta \right) > 0$

\int _{0}^{\infty } \frac{e^{-\alpha x}}{(\operatorname {sh} x)^{\beta }} d x = 2^{\beta -1} \mathrm{~ B}\left(\frac{\alpha +\beta }{2}, 1-\beta \right)

(4)

for $\mathrm{Re}\left(\beta \right) > -1, \mathrm{Re}\left(\alpha +\beta \right) < 0, \mathrm{Re}\left(\alpha -\beta \right) > 0$

\int _{0}^{\infty }(\operatorname {sh} x)^{\beta } \operatorname {sh} \alpha x d x = 2^{-\beta -2}\left[\mathrm{~ B}\left(\beta +1,-\frac{\alpha +\beta }{2}\right)-\mathrm{B}\left(\beta +1, \frac{\alpha -\beta }{2}\right)\right]

Problem 24.9 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Prove that for $\mathrm{Re}\left(\alpha \right) > 0, \mathrm{Re}\left(\beta \right) > 0$

\int _{-\pi / 4}^{\pi / 4}(\cos \varphi -\sin \varphi )^{2 \alpha -1}(\cos \varphi +\sin \varphi )^{2 \beta -1} d \varphi = 2^{\alpha +\beta -2} \mathrm{~ B}(\alpha , \beta )

Problem 24.10 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 24 "Integrals Reducible to the Gamma Function"

Prove that for $0 < \mathrm{Re}\left(\alpha \right) < 1$ the formula holds

\int _{-\pi / 4}^{\pi / 4}\left(\frac{\cos \varphi -\sin \varphi }{\cos \varphi +\sin \varphi }\right)^{2 \alpha -1} d \varphi = \frac{\pi }{2 \sin \pi \alpha }

References

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