Integrals of Gradshteyn and Ryzhik: 6.161 – Mellin Transforms of Theta Functions with Respect to the Lattice Parameter

We use definitions of the theta functions, shown below, from GR. Note that there is no standard notation for the theta functions.

\(z = \) argument, \(\tau = \) lattice parameter (\(\mathfrak{I}(\tau) \gt 0\)), and
\(q = \mathrm{e}^{i\pi \tau}\) (\(|q| \lt 1\))

\begin{align}
\tag{1a}
\label{eq:theta1e}
\theta_{1}(z|\tau) &= \theta_{1}(z,q) = 2 \sum_{n=0}^{\infty} (-1)^{n} q^{(n+1/2)^{2}} \sin[(2n+1)z] \\
\tag{1b}
\label{eq:theta1t}
&= -i \sum_{n=-\infty}^{\infty} (-1)^{n} q^{(n+1/2)^{2}} \mathrm{e}^{i(2n+1)z}
\end{align}

\begin{align}
\tag{2a}
\label{eq:theta2e}
\theta_{2}(z|\tau) &= \theta_{2}(z,q) = 2 \sum_{n=0}^{\infty} q^{(n+1/2)^{2}} \cos[(2n+1)z] \\
\tag{2b}
\label{eq:theta2t}
&= \sum_{n=-\infty}^{\infty} q^{(n+1/2)^{2}} \mathrm{e}^{i(2n+1)z}
\end{align}

\begin{align}
\tag{3a}
\label{eq:theta3e}
\theta_{3}(z|\tau) &= \theta_{3}(z,q) = 1 + 2 \sum_{n=1}^{\infty} q^{n^{2}} \cos(2nz) \\
\tag{3b}
\label{eq:theta3t}
&= \sum_{n=-\infty}^{\infty} q^{n^{2}} \mathrm{e}^{i2nz}
\end{align}

\begin{align}
\tag{4a}
\label{eq:theta4e}
\theta_{4}(z|\tau) &= \theta_{4}(z,q) = 1 + 2 \sum_{n=1}^{\infty} q^{n^{2}} \cos(2nz) \\
\tag{4b}
\label{eq:theta4t}
&= \sum_{n=-\infty}^{\infty} q^{n^{2}} \mathrm{e}^{i2nz}
\end{align}

6.161.1

\begin{equation}
\int_{0}^{\infty} x^{s-1} \theta_{2}(0|ix^2) dx
= 2\int_{0}^{\infty} x^{s-1} \sum_{n=0}^{\infty} \mathrm{e}^{-\pi x^{2}(n+1/2)^2} dx
\end{equation}

\begin{align}
\int_{0}^{\infty} x^{s-1} \mathrm{e}^{-\pi x^{2}(n+1/2)^2} dx
&= \frac{1}{2(\pi (n+1/2)^2)^{s/2}} \int_{0}^{\infty} y^{-1+s/2} \mathrm{e}^{-y} dy \\
&= \frac{1}{2(\pi (n+1/2)^2)^{s/2}} \Gamma(s/2)
\end{align}

We used the substitution \(y = \pi x^{2}(n+1/2)^2\).

Now we have
\begin{align}
\int_{0}^{\infty} x^{s-1} \theta_{2}(0|ix^2) dx &=
\frac{1}{\pi ^{s/2}} \Gamma(s/2) \sum_{n=0}^{\infty} \frac{1}{(n+1/2)^s} \\
&= \frac{2^s}{\pi ^{s/2}} \Gamma(s/2) \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s} \\
&= \frac{2^s}{\pi ^{s/2}} \Gamma(s/2) (1-2^{-s}) \zeta(s)
\end{align}

6.161.2

\begin{align}
\int_{0}^{\infty} x^{s-1} [\theta_{3}(0|ix^2) \,- 1] dx
&= \int_{0}^{\infty} x^{s-1} \left(\left[1 + 2\sum_{n=1}^{\infty} \mathrm{e}^{-\pi x^{2} n^2} \right] -1 \right) dx \\
&= 2\sum_{n=1}^{\infty} \int_{0}^{\infty} x^{s-1} \mathrm{e}^{-\pi x^{2} n^2} dx \\
&= 2\sum_{n=1}^{\infty} \frac{\Gamma(s/2)}{2\pi^{s/2} n^{s}} \\
&= \frac{\Gamma(s/2)}{\pi^{s/2}} \sum_{n=1}^{\infty} \frac{1}{n^{s}} \\
&= \frac{\Gamma(s/2)}{\pi^{s/2}} \zeta(s)
\end{align}

We used the substitution \(y = \pi x^{2} n^2\) and the same work as above to evaluate the integral.

6.161.3

\begin{align}
\int_{0}^{\infty} x^{s-1} [1 – \theta_{4}(0|ix^2)] dx
&= \int_{0}^{\infty} x^{s-1} \left(1 – \left[1 + 2\sum_{n=1}^{\infty} (-1)^{n} \mathrm{e}^{-\pi x^{2} n^2} \right] \right) dx \\
&= -2\sum_{n=1}^{\infty} (-1)^{n} \int_{0}^{\infty} x^{s-1} \mathrm{e}^{-\pi x^{2} n^2} dx \\
&= -2\sum_{n=1}^{\infty} (-1)^{n} \frac{\Gamma(s/2)}{2\pi^{s/2} n^{s}} \\
&= \frac{\Gamma(s/2)}{\pi^{s/2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{s}} \\
&= \frac{\Gamma(s/2)}{\pi^{s/2}} (1 – 2^{1-s}) \zeta(s)
\end{align}

We used the substitution \(y = \pi x^{2} n^2\) and the same work as above to evaluate the integral.

6.161.4

\begin{align}
\int_{0}^{\infty} x^{s-1} [\theta_{4}(0|ix^2) + \theta_{2}(0|ix^2) \,- \theta_{3}(0|ix^2)] dx
&= \int_{0}^{\infty} x^{s-1} ([\theta_{4}(0|ix^2) \,-1] + \theta_{2}(0|ix^2) \,- [\theta_{3}(0|ix^2) \,-1]) dx \\
&= \frac{\Gamma(s/2)}{\pi^{s/2}} (2^{1-s} – 1) \zeta(s)
+ \frac{2^s}{\pi ^{s/2}} \Gamma(s/2) (1-2^{-s}) \zeta(s)
\,- \frac{\Gamma(s/2)}{\pi^{s/2}} \zeta(s) \\
&= -\frac{\Gamma(s/2)}{\pi^{s/2}} \zeta(s) [1 – 2^{1-s} – 2^{-s} + 1 + 1] \\
&= -\frac{\Gamma(s/2)}{\pi^{s/2}} \zeta(s) (2^{-s} – 1)(2^{1-s} – 1)
\end{align}
Here we used the previous 3 results.

All Riemann zeta function expressions can be found here.

Evaluate \(\int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} dx\)

How to evaluate
\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} dx
\end{equation}
was a question posed at Mathematics Stack Exchange. Here is my solution to a more general problem.

We consider the more general integral
\begin{align}
\int\limits_{0}^{\infty} \frac{x^{n}}{\mathrm{e}^{ax}-1} dx
&= \int\limits_{0}^{\infty} \frac{\mathrm{e}^{-ax} x^{n}}{1-\mathrm{e}^{-ax}} dx \\
&= \int\limits_{0}^{\infty} \mathrm{e}^{-ax} x^{n} \sum\limits_{k = 0}^{\infty} \mathrm{e}^{-kax} dx \\
&= \sum\limits_{k = 0}^{\infty} \int\limits_{0}^{\infty} x^{n} \mathrm{e}^{-(a+ak)x} dx
\end{align}

To evaluate the integral, let \((a+ak)x = z\)
\begin{equation}
\int\limits_{0}^{\infty} x^{n} \mathrm{e}^{-(a+ak)x} dx
= \frac{1}{a^{n+1}} \frac{1}{(1+k)^{n+1}} \int\limits_{0}^{\infty} z^{n} \mathrm{e}^{-z} dz
= \frac{1}{a^{n+1}} \frac{1}{(1+k)^{n+1}} \Gamma(n+1)
\end{equation}

Now we have
\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{n}}{\mathrm{e}^{ax}-1} dx
= \frac{1}{a^{n+1}} \Gamma(n+1) \sum\limits_{k = 1}^{\infty} \frac{1}{k^{n+1}}
= \frac{1}{a^{n+1}} \Gamma(n+1) \zeta(n+1)
\end{equation}

The original problem is
\begin{equation}
\int\limits_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} dx
= \Gamma(4) \zeta(4) = 6\frac{\pi^{4}}{90} = \frac{\pi^{4}}{15}
\end{equation}

Evaluate the Integral \(\int_{0}^{\infty} \frac{\ln(x)}{1+e^{ax}} \mathrm{d}x\)

How to evaluate

\begin{equation}
I = \int\limits_{0}^{\infty} \frac{\ln(x)}{1+e^{ax}} \mathrm{d}x
\tag{1}
\label{eq:161008-1}
\end{equation}
was a question posed at Mathematics Stack Exchange. As of the time of writing this post, there was a good solution posted there. Here is my solution.

Let
\begin{align}
I_{1} &= \int\limits_{0}^{\infty} \frac{x^{b}}{1+e^{ax}} \mathrm{d}x = \int\limits_{0}^{\infty} \frac{x^{b}e^{-ax}}{1+e^{-ax}} \mathrm{d}x \\
&= \sum\limits_{n=0}^{\infty} (-1)^{n} \int\limits_{0}^{\infty} x^{b} e^{-(a+na)x} \mathrm{d}x
\end{align}

We designate the last integral on the right as \(I_{2}\) and make the substitution \(y=(a+na)x\)
\begin{align}
I_{2} &= \int\limits_{0}^{\infty} x^{b} e^{-(a+na)x} \mathrm{d}x \\
&= \frac{1}{(a+na)^{b+1}} \int\limits_{0}^{\infty} y^{b} e^{-y} \mathrm{d}y \\
&= \frac{\Gamma(b+1)}{(a+na)^{b+1}} \\
&= \frac{\Gamma(b+1)}{a^{b+1}} \frac{1}{(n+1)^{b+1}}
\end{align}

Now \(I_{1}\) becomes
\begin{equation}
I_{1} = \frac{\Gamma(b+1)}{a^{b+1}} \sum\limits_{n=0}^{\infty} (-1)^{n} \frac{1}{(n+1)^{b+1}} = \frac{\Gamma(b+1)}{a^{b+1}} \eta(b+1)
\end{equation}
and we have
\begin{align}
I &= \lim_{b \to 0} \frac{\partial I_{1}}{\partial b} = \lim_{b \to 0} \int\limits_{0}^{\infty} \frac{x^{b} \ln(x)}{1+e^{ax}} \mathrm{d}x \\
&= \lim_{b \to 0} \frac{\partial}{\partial b} \frac{\Gamma(b+1)\eta(b+1)}{a^{b+1}} \\
&= \lim_{b \to 0} \frac{a^{b+1} \Big[ \Gamma(b+1)\eta^{\prime}(b+1) + \Gamma^{\prime}(b+1)\eta(b+1) \Big] – \Gamma(b+1)\eta(b+1)a^{b+1}\ln(a)}{\left(a^{b+1}\right)^{2}} \\
&= \lim_{b \to 0} \frac{\Gamma(b+1)\eta^{\prime}(b+1) + \Gamma^{\prime}(b+1)\eta(b+1) – \Gamma(b+1)\eta(b+1)\ln(a)}{a^{b+1}} \\
\tag{a}
&= \frac{1}{a} \left(\Big[\gamma \ln(2) – \frac{1}{2} \ln^{2}(2)\Big] -\gamma \ln(2) – \ln(2)\ln(a) \right) \\
&= -\frac{\ln(2)}{a} \left(\frac{1}{2} \ln(2) + \ln(a) \right)
\end{align}

In step (a) we have
\begin{align}
\lim_{b \to 0} \Gamma^{\prime}(b+1)\eta(b+1) &= \lim_{b \to 0} \Gamma^(b+1)\psi(b+1)\eta(b+1) \\
&= \Gamma(1)\psi(1)\eta(1) \\
&= -\gamma \ln(2)
\end{align}
and
\begin{align}
\lim_{b \to 0} \Gamma(b+1)\eta^{\prime}(b+1) &= \lim_{s \to 1} \eta^{\prime}(s) \\
&= \lim_{s \to 1} \sum\limits_{n=0}^{\infty} (-1)^{n} \frac{\ln(n)}{n^{s}} \\
&= \gamma \ln(2) – \frac{1}{2} \ln^{2}(2)
\end{align}
See here for a proof of this result.

Notes:

  1. \(\Gamma(z)\) is the Gamma function.
  2. \(\eta(s)\) is the Dirichlet eta function.
  3. \(\zeta(s)\) is the Riemann zeta function.
  4. \(\psi(z)\) is the digamma function.
  5. \(\gamma\) is the Euler-Mascheroni constant.

Integrals of products of Hurwitz zeta functions via Feynman parametrization and two double sums of Riemann zeta functions by M. A. Shpot, R. B. Paris

We consider two integrals over \(x \in [0,1]\) involving products of the function \(\zeta_{1}(a,x) \equiv \zeta(a,x) − x^{−a}\), where \(\zeta(a,x)\) is the Hurwitz zeta function, given by
\begin{equation}
\int\limits_{0}^{1} \zeta_{1}(a,x) \zeta_{1}(b,x) \mathrm{d}x
\end{equation}
and
\begin{equation}
\int\limits_{0}^{1} \zeta_{1}(a,x) \zeta_{1}(b,1-x) \mathrm{d}x
\end{equation}

when \(\Re (a,b) \gt 1\). These integrals have been investigated recently in [23]; here we provide an alternative derivation by application of Feynman parametrization. We also discuss a moment integral and the evaluation of two doubly infinite sums containing the Riemann zeta function \(\zeta(x)\) and two free parameters a and b. The limiting forms of these sums when \(a+b\) takes on integer values are considered.

The paper is available 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}

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

Integrate \(\int_{0}^{1} \frac{\mathrm{ln}^{3}(x)}{1+x} \mathrm{d} x\)

The following integral was a question on Mathematics Stack Exchange

\begin{equation}
\int\limits_{0}^{1} \frac{\mathrm{ln}^{3}(x)}{1+x} \mathrm{d} x
\label{eq:160812a1}
\tag{1}
\end{equation}

We begin with the substitution \(x=\mathrm{e}^{-y}\) so our integral becomes
\begin{equation}
-\int\limits_{0}^{\infty} \frac{y^{3}}{1+\mathrm{e}^{\,y}} \mathrm{d} y
\label{eq:160812a2}
\tag{2}
\end{equation}

This integral is a Mellin transform of the function
\begin{equation}
f(y) = \frac{1}{1+\mathrm{e}^{\,y}}
\label{eq:160812a3}
\tag{3}
\end{equation}
where the Mellin transform is defined as
\begin{equation}
\mathcal{M}[f(x)](s) = \int\limits_{0}^{\infty} x^{s-1} f(x) \mathrm{d} x
\label{eq:160812a4}
\tag{4}
\end{equation}

From Tables of Integral Transforms (Bateman Manuscript) Volume 1, we have
\begin{equation}
\mathcal{M}[(\mathrm{e}^{\alpha x} + 1)^{-1}](s) = \frac{1}{\alpha^{s}} \Gamma(s) (1-2^{1-s}) \zeta(s)
\label{eq:160812a5}
\tag{5}
\end{equation}

With \(s = 4\) and \(\alpha = 1\) we have
\begin{equation}
-\int\limits_{0}^{\infty} \frac{y^{3}}{1+\mathrm{e}^{\,y}} \mathrm{d} y = -\Gamma(4) (1-2^{1-4}) \zeta(4) = -\frac{7\pi^{4}}{120}
\label{eq:160812a6}
\tag{6}
\end{equation}

Thus
\begin{equation}
\int\limits_{0}^{1} \frac{\mathrm{ln}^{3}(x)}{1+x} \mathrm{d} x = -\frac{7\pi^{4}}{120}
\label{eq:160812a7}
\tag{7}
\end{equation}

Addendum

We can rewrite \eqref{eq:160812a5} as
\begin{align}
\mathcal{M}[(\mathrm{e}^{\alpha x} + 1)^{-1}](s) & = \frac{1}{\alpha^{s}} \Gamma(s) (1-2^{1-s}) \zeta(s) \\
& = \frac{1}{\alpha^{s}} \Gamma(s) \eta(s)
\label{eq:160812a8}
\tag{8}
\end{align}
where
\begin{equation}
\eta(s) = (1-2^{1-s}) \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}
\end{equation}
is the Dirichlet eta function, also known as the alternating Riemann zeta function.

Note that for Wolfram Alpha, \(\eta(s)\) defaults to the Dedekind eta function. To obtain the Dirichlet eta function type “dirichlet eta(s)”.