A new proof of the fundamental two-term transformation for the series \(_{3}\mathrm{F}_{2}(1)\) due to Thomae by Rathie

The aim of this short note is to provide a very simple proof for obtaining the fundamental two-term transformation for the series \(_{3}\mathrm{F}_{2}(1)\) due to Thomae.

The paper can be found here.

Integrate \(\int_{0}^{\infty} \frac{x^{4}}{(\mathrm{e}^{x}-1)^{2}} \mathrm{d} x \)

How to evaluate
\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{4}}{(\mathrm{e}^{x}-1)^{2}} \mathrm{d} x
\end{equation}
was a question on Mathematics Stack Exchange which for some inexplicable reason has been closed. Here is another solution.

Let
\begin{equation}
f(x) = \frac{1}{(\mathrm{e}^{x}-1)^{2}}
\end{equation}
Then the Mellin transform of the function is
\begin{equation}
\mathcal{M}[f(x)](s) = \int\limits_{0}^{\infty} \frac{x^{s-1}}{(\mathrm{e}^{x}-1)^{2}} \mathrm{d} x
= \Gamma(s)[\zeta(s-1) – \zeta(s)]
\end{equation}
For \(s=5\), we have
\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{4}}{(\mathrm{e}^{x}-1)^{2}} \mathrm{d} x = \Gamma(5)[\zeta(4) – \zeta(5)]
= \frac{4\pi^{4}}{15} – 24\zeta(5)
\end{equation}

Integrate \(\int_{0}^{\infty} (\sqrt{x^{4}+a^{4}} + \sqrt{x^{4}+b^{4}})^{-1} \mathrm{d} x\)

How to evaluate
\begin{equation}
\int\limits_{0}^{\infty} \frac{1}{\sqrt{x^{4}+a^{4}} + \sqrt{x^{4}+b^{4}}} \mathrm{d} x
\label{eq:160824-1}
\tag{1}
\end{equation}

was a question on Mathematics Stack Exchange. One of the answers outlined a way to convert the integral into beta functions but did not work out the solution. To do so requires a clever trick to ensure convergence of the integral, which is my reason for posting a full solution here.

We begin by multiplying the integrand by
\begin{equation}
\frac{\sqrt{x^{4}+a^{4}} – \sqrt{x^{4}+b^{4}}}{\sqrt{x^{4}+a^{4}} – \sqrt{x^{4}+b^{4}}}
\end{equation}
so that we now have
\begin{equation}
\frac{1}{a^{4}-b^{4}} \int\limits_{0}^{\infty} \sqrt{x^{4}+a^{4}} – \sqrt{x^{4}+b^{4}} \, \mathrm{d} x
\label{eq:160824-2}
\tag{2}
\end{equation}

The problem here is that
\begin{equation}
\int\limits_{0}^{\infty} \sqrt{x^{4}+a^{4}} \, \mathrm{d} x
\end{equation}
does not converge. Here is where we invoke the clever trick I referred to. Let us consider
\begin{equation}
\int\limits_{0}^{\infty} (x^{4}+a^{4})^{p} \mathrm{d} x
\label{eq:160824-3}
\tag{3}
\end{equation}
Making the successive substitutions \(x^{4} = a^{4}y^{4}\) and \(y^{4} = z\) yields
\begin{align}
\int\limits_{0}^{\infty} (x^{4}+a^{4})^{p} \mathrm{d} x & = \frac{a^{4p+1}}{4} \int\limits_{0}^{\infty} z^{-3/4} (z+1)^{p} \mathrm{d} z \\
& = \frac{a^{4p+1}}{4} \mathrm{B}\left(\frac{1}{4},\frac{-1}{4}-p\right)
\label{eq:160824-4}
\tag{4}
\end{align}

Putting it all together and letting \(p = 1/2\), we have
\begin{align}
\int\limits_{0}^{\infty} \frac{1}{\sqrt{x^{4}+a^{4}} + \sqrt{x^{4}+b^{4}}} \mathrm{d} x & = \frac{1}{a^{4}-b^{4}} \int\limits_{0}^{\infty} \sqrt{x^{4}+a^{4}} – \sqrt{x^{4}+b^{4}} \mathrm{d} x \\
& = \frac{1}{4} \frac{a^{3}-b^{3}}{a^{4}-b^{4}} \mathrm{B}\left(\frac{1}{4},\frac{-3}{4}\right) \\
& \approx \frac{a^{3}-b^{3}}{a^{4}-b^{4}} 1.23605
\end{align}

Computing hypergeometric functions rigorously

“Computing hypergeometric functions rigorously” is the title of a recent paper by Fredrik Johansson. Here is the abstract:

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function pFq and computation of high-order parameter derivatives.

H/T Analysis Fact (Twitter link)

Convergence of \(\int_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x\)

The question for what values of \(p\) does the improper integral
\begin{equation}
\int\limits_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x
\end{equation}
converge was asked at Mathematics Stack Exchange.

My solution was to note that this integral can be expressed in terms of the upper incomplete Gamma function
\begin{equation}
\Gamma(a,x) = \int\limits_{x}^{\infty} \mathrm{e}^{-z} z^{a-1} \mathrm{d} z
\end{equation}
Thus
\begin{equation}
\int\limits_{1}^{\infty} \mathrm{e}^{-x} x^{p} \mathrm{d} x = \Gamma(1+p,1)
\end{equation}
As noted here, the upper incomplete Gamma function, \(\Gamma(a,x)\) is an entire
function for all \(a\) when \(x \ne 0\). Thus the integral converges for all values
of \(p\).

Integrate \(\int_{0}^{\infty} \frac{x}{\mathrm{e}^{x}+1} \mathrm{d} x\)

How to evaluate
\begin{equation}
\int\limits_{0}^{\infty} \frac{x}{\mathrm{e}^{x}+1} \mathrm{d} x
\end{equation}
was a question on Mathematics Stack Exchange. The original post asked for a contour integral solution and there is a good one among the answers. However, this integral can be evaluated trivially by recognizing an integral definition of the Dirichlet eta function or use of the Mellin transform.

We have
\begin{equation}
\eta(s) = \frac{1}{\Gamma(s)}\int\limits_{0}^{\infty} \frac{x^{s-1}}{\mathrm{e}^{x}+1} \mathrm{d} x
\end{equation}
thus our integral is
\begin{equation}
\int\limits_{0}^{\infty} \frac{x}{\mathrm{e}^{x}+1} \mathrm{d} x = \eta(2)\Gamma(2) = \frac{\pi^{2}}{12}
\end{equation}

Also
\begin{align}
\mathcal{M}\left[(\mathrm{e}^{ax}+1)^{-1}\right](s) & = \int\limits_{0}^{\infty} \frac{x^{s-1}}{\mathrm{e}^{ax}+1} \mathrm{d} x \\
& = a^{-s}\Gamma(s)(1-2^{1-s})\zeta(s) \\
& = a^{-s}\Gamma(s)\eta(s)
\end{align}
With \(a=1\) and \(s=2\) we arrive at our result.

Laplace Transform of the Gudermannian Function

How to find the Laplace transform of the Gudermannian function was a question on Mathematics Stack Exchange. Here is my solution.

\begin{equation}
\mathcal{L}[\mathrm{gd}(x)](s) = \int\limits_{0}^{\infty} \mathrm{e}^{-sx}\mathrm{gd}(x)\mathrm{d} x
\end{equation}
Integration by parts yields
\begin{align}
\int\limits_{0}^{\infty} \mathrm{e}^{-sx}\mathrm{gd}(x)\mathrm{d} x & =
\frac{-1}{s}\mathrm{gd}(x)\mathrm{e}^{-sx}\big|_{0}^{\infty} +
\frac{1}{s}\int\limits_{0}^{\infty} \mathrm{e}^{-sx}\mathrm{sech}(x)\mathrm{d} x \\
& = 0 + \frac{1}{s}\mathcal{L}[\mathrm{sech}(x)](s) \\
& = \frac{1}{2s}\left[\psi\left(\frac{s+3}{4}\right) – \psi\left(\frac{s+1}{4}\right) \right]
\end{align}
for \(Re(s) > 0\) due to evaluating the limit at \(x = \infty\), while \(Re(s) > -1\) for the Laplace transform of the hyperbolic secant.

Notes:

1. \(\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{gd}(x) = \mathrm{sech}(x)\)
2. \(\mathrm{gd}(0) = 0\,\) and \(\,\mathrm{gd}(\infty) = \frac{\pi}{2}\)
3. \(\psi(s)\) is the digamma function.

Integrate \(\int_{0}^{\infty} \frac{x}{\sqrt{\mathrm{e}^{x}-1}} \mathrm{d} x\)

How to evaluate
\begin{equation}
\int\limits_{0}^{\infty} \frac{x}{\sqrt{\mathrm{e}^{x}-1}} \mathrm{d} x
\end{equation}
was a question on Mathematics Stack Exchange. Here is another solution method.

Let \(z=\mathrm{e}^{x}-1\), so that we have
\begin{equation}
\int\limits_{0}^{\infty} \frac{\mathrm{ln}(z+1)}{\sqrt{z}}\frac{1}{z+1} \mathrm{d} z
\end{equation}

Let us consider
\begin{equation}
I(a) = \int\limits_{0}^{\infty} \frac{(z+1)^{a}}{\sqrt{z}} \mathrm{d} z = \mathrm{B}\left(\frac{1}{2}, -\frac{1}{2}-a\right)
= \frac{\Gamma\left(\frac{1}{2}\right) \Gamma\left(-\frac{1}{2}-a\right)}{\Gamma(-a)}
\end{equation}
so that
\begin{equation}
\lim_{a \to -1} \frac{\partial I(a)}{\partial a} = \int\limits_{0}^{\infty} \frac{\mathrm{ln}(z+1)}{\sqrt{z}}\frac{1}{z+1} \mathrm{d} z =
\int\limits_{0}^{\infty} \frac{x}{\sqrt{\mathrm{e}^{x}-1}} \mathrm{d} x
\end{equation}

Then,
\begin{equation}
\frac{\partial I(a)}{\partial a} = \Gamma\left(\frac{1}{2}\right)\left[\frac{-\Gamma(-a)\Gamma\left(-\frac{1}{2}-a\right)\psi^{0}\left(-\frac{1}{2}-a\right) + \Gamma\left(-\frac{1}{2}-a\right)\Gamma(-a)\psi^{0}(-a)}{\Gamma(-a)\Gamma(-a)} \right]
\end{equation}

\begin{align}
\lim_{a \to -1} \frac{\partial I(a)}{\partial a} & = \frac{-\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma(1)}
\left[\psi^{0}\left(\frac{1}{2}\right) – \psi^{0}(1)\right] \\
& = -\pi[(-\gamma-\mathrm{ln}4) -(- \gamma)] \\
& = \pi\mathrm{ln}4 \\
& = \int\limits_{0}^{\infty} \frac{x}{\sqrt{\mathrm{e}^{x}-1}} \mathrm{d} x
\end{align}

Integrals from Blagouchine’s Malmsten Integral Paper

Here I evaluate problem 18a from page 43 from Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results by Iaroslav V. Blagouchine as well as a bonus integral. The integral in question is

\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{a}\mathrm{ln}(x)}{\mathrm{e}^{bx}-1} \mathrm{d} x
\label{eq:160817-1}
\tag{1}
\end{equation}

We will evaluate this integral using two methods. First, taking the hint given by Blagouchine, we expand the function
\begin{equation}
f(x) = \frac{1}{\mathrm{e}^{bx}-1}
\label{eq:160817-2}
\tag{2}
\end{equation}
and then integrate term by term.

\begin{equation}
f(x) = \frac{1}{\mathrm{e}^{bx}-1} = \frac{\mathrm{e}^{-bx}}{1-\mathrm{e}^{-bx}} = \sum\limits_{n=0}^{\infty} \mathrm{e}^{-(1+n)bx}
\label{eq:160817-3}
\tag{3}
\end{equation}

We commence with
\begin{equation}
I = \int\limits_{0}^{\infty} \frac{x^{a}}{\mathrm{e}^{bx}-1} \mathrm{d} x = \int\limits_{0}^{\infty} x^{a} \sum\limits_{n=0}^{\infty} \mathrm{e}^{-(1+n)bx} \mathrm{d} x
= \sum\limits_{n=0}^{\infty} \int\limits_{0}^{\infty} x^{a} \mathrm{e}^{-(1+n)bx} \mathrm{d} x
\label{eq:160817-4}
\tag{4}
\end{equation}

Making the substitution \(y = (1+n)bx\) yields
\begin{align}
I & = \sum\limits_{n=0}^{\infty} \frac{1}{(1+n)^{a+1}b^{a+1}} \int\limits_{0}^{\infty} y^{a} \mathrm{e}^{-y} \mathrm{d} y
= \frac{\Gamma(a+1)}{b^{\,a+1}} \sum\limits_{n=0}^{\infty} \frac{1}{(1+n)^{a+1}} \\
& = \frac{\Gamma(a+1)}{b^{\,a+1}} \sum\limits_{k=1}^{\infty} \frac{1}{k^{a+1}} \\
& = \frac{\Gamma(a+1)\zeta(a+1)}{b^{\,a+1}}
\label{eq:160817-5}
\tag{5}
\end{align}

So we have our first result
\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{a}}{\mathrm{e}^{bx}-1} \mathrm{d} x = \frac{\Gamma(a+1)\zeta(a+1)}{b^{\,a+1}}
\label{eq:160817-6}
\tag{6}
\end{equation}

We can evaluate this integral quite simply if we recognize that it can be written as a Mellin transform
\begin{equation}
\mathcal{M}[f(x)](s) = \int\limits_{0}^{\infty} \frac{x^{s-1}}{\mathrm{e}^{bx}-1} \mathrm{d} x = \frac{\Gamma(s)\zeta(s)}{b^{s}}
\label{eq:160817-7}
\tag{7}
\end{equation}
Letting \(s = a+1\) yields our result.

To evaluate the integral in \eqref{eq:160817-1} we differentiate equation \eqref{eq:160817-6} with respect to \(a\) and note that
\begin{equation}
\frac{d\Gamma(z)}{dz} = \Gamma(z)\psi^{(0)}(z)
\end{equation}

\begin{align}
\int\limits_{0}^{\infty} \frac{x^{a}\mathrm{ln}(x)}{\mathrm{e}^{bx}-1} \mathrm{d} x & = \frac{-\mathrm{ln}(b)}{b^{\,a+1}} \Gamma(a+1)\zeta(a+1) + \frac{\Gamma'(a+1)\zeta(a+1)}{b^{\,a+1}} + \frac{\Gamma(a+1)\zeta'(a+1)}{b^{\,a+1}} \\
& = \frac{\Gamma(a+1)}{b^{\,a+1}} \left[ \psi^{(0)}(a+1)\zeta(a+1) + \zeta'(a+1) – \zeta(a+1)\mathrm{ln}(b) \right]
\label{eq:160817-8}
\tag{8}
\end{align}

Note that \(\mathrm{Re}(a) > 0\).

Derivation of an Integral Expression of the Euler-Mascheroni Constant – Part 2

I derived the expression
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} \mathrm{ln}(x) \mathrm{d} x = -\gamma
\label{eq:1608161}
\tag{1}
\end{equation}
for the Euler-Mascheroni constant here. However, there is a far easier method that was fully derived in Advanced Integration Techniques by Zaid Alyafeai. I recommend this book to readers of this blog. It is free and contains many useful and interesting results.

We start with
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} x^t \mathrm{d} x = \Gamma(t+1)
\label{eq:1608162}
\tag{2}
\end{equation}
Differentiate with respect to \(t\)
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} x^t \mathrm{ln}(x) \mathrm{d} x = \frac{d\Gamma(t+1)}{dt} = \Gamma(t+1) \psi^{(0)}(t+1)
\label{eq:1608163}
\tag{3}
\end{equation}

Taking the limit of equation \eqref{eq:1608163}, \(t \to 0\) yields
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} \mathrm{ln}(x) \mathrm{d} x = \Gamma(1) \psi^{(0)}(1) = -\gamma
\label{eq:1608164}
\tag{4}
\end{equation}