Integrals of Rational Functions of Jacobi Elliptic Functions

Note that this post will be continuously updated as results are required for evaluating other integrals.

We use results from the relationships among squares of Jacobi elliptic functions.

Also, we use the numbering scheme of Handbook of Elliptic Integrals for Engineers and Scientists by Byrd and Friedman.

We begin with the definition of the incomplete elliptic integral of the third kind
\begin{equation}
\Pi(\phi,\alpha^{2},k) = \int \frac{du}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}(u,k)}
\end{equation}

and
\begin{equation}
\int du = u
\end{equation}
which we note here as we drop the differential and dependent variables in the work below.

BF 337.01

We begin with
\begin{align}
\int \frac{\alpha^{2}\mathrm{sn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} &= \int \frac{1 \,-\, 1 + \alpha^{2}\mathrm{sn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} \\
&= \int \frac{1}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} \,-\, \int \frac{1 \,-\, \alpha^{2}\mathrm{sn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} = \Pi \,-\, u
\end{align}
Rearranging yields
\begin{equation}
\int \frac{\mathrm{sn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} = \frac{\Pi \,-\, u}{\alpha^{2}}
\end{equation}

BF 338.01

\begin{align}
\int \frac{\mathrm{cn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} &= \int \frac{1}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} \,-\, \int \frac{\mathrm{sn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} \\
&= \Pi \,-\, \frac{\Pi \,-\, u}{\alpha^{2}} = \frac{1}{\alpha^{2}}\left[(\alpha^{2} \,-\, 1)\Pi + u \right]
\end{align}

BF 339.01

\begin{align}
\int \frac{\mathrm{dn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} &= \int \frac{1 \,-\, k^{2}\mathrm{sn}^{2}}{1 \,-\, \alpha^{2}\mathrm{sn}^{2}} \\
&= \Pi \,-\, \frac{k^{2}}{\alpha^{2}}(\Pi \,-\, u) = \frac{1}{\alpha^{2}}\left[(\alpha^{2} \,-\, k^{2})\Pi + k^{2}u \right]
\end{align}

Integrals of Jacobi Elliptic Functions Raised to the Power 4

We use combinations of the following to evaluate integrals in this post:

All results can be found in section 310 – 321 of Handbook of Elliptic Integrals for Engineers and Scientists by Byrd and Friedman. We use their equation numbering scheme.

BF 310.04

\begin{align}
\int \mathrm{sn}^{4} &= \int \mathrm{sn}^{2} (1 \,-\, \mathrm{cn}^{2}) = \int \mathrm{sn}^{2} \,-\, \int \mathrm{sn}^{2} \mathrm{cn}^{2} \\
&= \frac{1}{3k^{4}}\left[(k^{2} + 2)u \,-\, 2(k^{2} + 1)E + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{align}
We used GR 5.134.1 and BF 361.01.

BF 312.04

\begin{align}
\int \mathrm{cn}^{4} &= \int \mathrm{cn}^{2} (1 \,-\, \mathrm{sn}^{2}) = \int \mathrm{cn}^{2} \,-\, \int \mathrm{sn}^{2} \mathrm{cn}^{2} \\
&= \frac{1}{3k^{4}}[{k^{\prime}}^{2}(2 \,-\, 3k^{2})u + 2(2k^{2} \,-\, 1)E + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} ]
\end{align}
We used GR 5.134.2 and BF 361.01.

BF 314.04

\begin{align}
\int \mathrm{dn}^{4} &= \int \mathrm{dn}^{2} (1 \,-\, k^{2}\mathrm{sn}^{2}) = \int \mathrm{dn}^{2} \,-\, k^{2}\int \mathrm{sn}^{2} \mathrm{dn}^{2} \\
&= \frac{1}{3}[2(1 + {k^{\prime}}^{2})E \,-\, {k^{\prime}}^{2}u + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} ]
\end{align}
We used GR 5.134.3 and BF 361.02.

Continue reading “Integrals of Jacobi Elliptic Functions Raised to the Power 4”

Integrals of Gradshteyn and Ryzhik: 2.484 – Combinations of hyperbolic functions, exponentials, and powers

Preliminary Work

We use the following definition of the exponential integral function
\begin{equation}
\mathrm{Ei}(z) = \int_{-\infty}^{z} \frac{\mathrm{e}^{x}}{x}dx
\end{equation}
We also have,
\begin{equation}
\mathrm{Ei}(kz) = \int_{-\infty}^{kz} \frac{\mathrm{e}^{w}}{w}dw
\end{equation}
which can be obtained by letting \(w=kx\).

Additionally, we will require the following, which can be obtained via integration by parts
\begin{equation}
\label{eq:180831-1}
\tag{1}
\int \frac{\mathrm{e}^{kx}}{x^{2}}dx = -\,\frac{\mathrm{e}^{kx}}{x} + k\int \frac{\mathrm{e}^{kx}}{x} = k\,\mathrm{Ei}(kx)\,-\,\frac{\mathrm{e}^{kx}}{x}
\end{equation}

For numbers 1-4 below, \(a^{2} \ne b^{2}\), while for 5-10, \(a = b\).

2.484.1

\begin{align}
\int \frac{1}{x}\mathrm{e}^{ax}\sinh(bx)\,dx &= \frac{1}{2} \int \frac{1}{x}[\mathrm{e}^{(a+b)x} \,-\, \mathrm{e}^{(a-b)x}] \\
&= \frac{1}{2} \left(\mathrm{Ei}[(a+b)x] \,-\, \mathrm{Ei}[(a-b)x]\right)
\end{align}

2.484.2

\begin{align}
\int \frac{1}{x}\mathrm{e}^{ax}\cosh(bx)\,dx &= \frac{1}{2} \int \frac{1}{x}[\mathrm{e}^{(a+b)x} + \mathrm{e}^{(a-b)x}] \\
&= \frac{1}{2} \left(\mathrm{Ei}[(a+b)x] + \mathrm{Ei}[(a-b)x]\right)
\end{align}

Continue reading “Integrals of Gradshteyn and Ryzhik: 2.484 – Combinations of hyperbolic functions, exponentials, and powers”

Integrals of Pairs of Products of Squares of Jacobi Elliptic Functions

We use combinations of the following to evaluate integrals in this post:

The order of presentation here is done to avoid look ahead bias, thus avoiding circularity when referencing previous results.

All results can be found in section 361 of Handbook of Elliptic Integrals for Engineers and Scientists by Byrd and Friedman. We use their equation numbering scheme.

Note that this post will be continuously updated as results are required for evaluating other integrals.

BF 361.02

We begin with
\begin{equation}
\frac{\partial}{\partial u}\mathrm{cn} = -\,\mathrm{sn} \, \mathrm{dn}
\end{equation}
Squaring both sides, integrating by parts, and then making substitutions for \(\mathrm{cn}^{2}\), we have
\begin{align}
\int \mathrm{sn}^{2} \mathrm{dn}^{2} &= \int \left(\frac{\partial}{\partial u}\mathrm{cn}\right) \left(\frac{\partial}{\partial u}\mathrm{cn}\right) \\
&= -\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} + \int \mathrm{cn}^{2} \mathrm{dn}^{2} – k^{2}\int \mathrm{cn}^{2} \mathrm{sn}^{2} \\
&= -\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} + \int (1 – \,\mathrm{sn}^{2})\,\mathrm{dn}^{2} – \int (\mathrm{dn}^{2} – {k^{\prime}}^{2} ) \mathrm{sn}^{2} \\
&= \frac{1}{3 k^{2}}\left[{k^{\prime}}^{2}u + (2k^{2} – 1)E \,-\, k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{align}

BF 361.01

From BF 361.02 we have
\begin{equation}
\int \mathrm{sn}^{2} ({k^{\prime}}^{2} + k^{2}\mathrm{cn}^{2}) = \frac{1}{3 k^{2}}\left[{k^{\prime}}^{2}u + (2k^{2} – 1)E \,-\, k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{equation}
Rearranging yields
\begin{equation}
\int \mathrm{sn}^{2} \mathrm{cn}^{2} = \frac{1}{3 k^{4}}\left[(2 -\,k^{2})E \,-\,2{k^{\prime}}^{2}u \,-\, k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{equation}

BF 361.03

From BF 361.02 we have
\begin{equation}
\int (1 \,-\,\mathrm{cn}^{2}) \mathrm{dn}^{2} = \frac{1}{3 k^{2}}\left[{k^{\prime}}^{2}u + (2k^{2} – 1)E \,-\, k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{equation}
Rearranging yields
\begin{equation}
\int \mathrm{dn}^{2} \mathrm{cn}^{2} = \frac{1}{3 k^{2}}\left[(1+k^{2})E \,-\,{k^{\prime}}^{2}u + k^{2}\,\mathrm{cn} \,\mathrm{dn}\,\mathrm{sn} \right]
\end{equation}

Continue reading “Integrals of Pairs of Products of Squares of Jacobi Elliptic Functions”

Integrals of Gradshteyn and Ryzhik: 5.132, 5.133, 5.135 – Integrals of Jacobi Elliptic Functions

In this post we evaluate integrals of the 12 Jacobi elliptic functions. For most of these integrals, there are multiple expressions which are all equal upto a constant. We use a mixed strategy. Sometimes multiple results will be obtained from the integral directly using different substitutions. Other times a result will be shown to be the same as other results that appear in GR. This latter strategy usually consists of converting between log expressions and inverse trig and hyperbolic expressions.

Also, extensive use is made of derivatives and relationships between squares of the Jacobi elliptic functions. All equation numbers are references to the latter.

When there are multiple GR solutions, each is enclosed by a blue box.

For the sake of clarity, we drop the argument \(u\) and the modulus \(k\) from all expressions,
thus \(\mathrm{sn}(u,k) = \mathrm{sn}\). Additionally, we drop the \(du\) to avoid confusion with the function \(\mathrm{dn}\).

5.132.1

\begin{align}
\int \mathrm{ns} &= \int \frac{1}{\mathrm{sn}} = \int \frac{{k^{\prime}}^{2}\mathrm{sn}}{{k^{\prime}}^{2}\mathrm{sn}^{2}}
= {k^{\prime}}^{2} \int \frac{\mathrm{sn}}{\mathrm{dn}^{2} – \mathrm{cn}^{2}}
= {k^{\prime}}^{2} \int \frac{\mathrm{sn}}{\mathrm{cn}^{2}} \frac{1}{(\mathrm{dn}/\mathrm{cn})^{2} – 1} \\
&= \int \frac{dw}{w^{2}-1} = \frac{1}{2} \ln\left(\frac{1-w}{1+w}\right)
= \frac{1}{2} \ln\left(\frac{\mathrm{cn} – \mathrm{dn}}{\mathrm{cn} + \mathrm{dn}}\right)
\end{align}
We used equation 17 and the substitution \(w=\mathrm{dn}/\mathrm{cn} = \mathrm{dc}\).

To obtain the form of the solution in GR, we expand the logarithm and again use equation 17 (factored).
\begin{align}
\frac{1}{2} \ln\left(\frac{\mathrm{cn} – \mathrm{dn}}{\mathrm{cn} + \mathrm{dn}}\right) &=
\frac{1}{2} [\ln(\mathrm{cn} – \mathrm{dn}) – \ln(\mathrm{cn} + \mathrm{dn})] \\
&= \frac{1}{2} \ln(k^{2}-1) + \ln(\mathrm{sn})- \frac{1}{2} \ln(\mathrm{cn} + \mathrm{dn})
– \frac{1}{2} \ln(\mathrm{cn} + \mathrm{dn}) \\
&= \bbox[5px,border:2px solid blue] {\ln\left(\frac{\mathrm{sn}}{\mathrm{cn} + \mathrm{dn}}\right)} + \frac{1}{2} \ln(k^{2}-1)
\end{align}
the first term is the solution given in GR, while the second term is a constant of integration.

Using equation 17 again, we have
\begin{align}
\frac{1}{2} \ln\left(\frac{\mathrm{cn} – \mathrm{dn}}{\mathrm{cn} + \mathrm{dn}}\right) &=
\frac{1}{2} \ln\left(-\frac{(\mathrm{dn}\,-\mathrm{cn})^2}{\mathrm{dn}^{2}-\mathrm{cn}^{2}} \right)
= \frac{1}{2} \ln\left(-\frac{(\mathrm{dn}\,-\mathrm{cn})^2}{{k^{\prime}}^{2}\mathrm{sn}^{2}} \right) \\
&= \bbox[5px,border:2px solid blue] { \ln\left(\frac{\mathrm{dn}\,-\mathrm{cn}}{\mathrm{sn}} \right) } \,- \ln k^{\prime} + \frac{1}{2} \ln(-1)
\end{align}

Continue reading “Integrals of Gradshteyn and Ryzhik: 5.132, 5.133, 5.135 – Integrals of Jacobi Elliptic Functions”

Integrals of Gradshteyn and Ryzhik: 5.134 – Integrals of Squares of Jacobi Elliptic Functions – Revised

In this post, we make use of results from Relationships Between Squares of Jacobi Elliptic Functions, Derivatives of Jacobi Elliptic Functions, and Derivatives of Jacobi Elliptic Functions – Part 2. We will refer to equations in the first and third of these posts with i.e 12s and 12d2.

Note that only the first three integrals appear explicitly in Gradshteyn and Ryzhik. The other 9 appear implicitly as part of solutions of elliptic integrals. Thus we establish results here and label them for use in future posts. These 9 integrals can be found in Handbook of Elliptic Integrals for Engineers and Scientists by Byrd and Friedman, sections 310 – 321.

Addendum – Previously, we used the integrals of products of pairs of Jacobi elliptic functions to evaluate some of the integrals in this post. This method was somewhat dubious as it was due to working backwards from known results. We have revised the work so that we only require knowledge of derivatives of single Jacobi elliptic functions and thus no need to “look ahead”. We now do this below and rearrange the order so that it is clear how all integrals were evaluated.

GR 5.134.3

\begin{equation}
\int \mathrm{dn}^{2}(u,k)\,du = \mathrm{E}(\mathrm{am}\,u,k) = \mathrm{E}(u)
\end{equation}
this is a definition of the incomplete elliptic integral of the second kind. \(\mathrm{am}\,u\) is the amplitude function.

GR 5.134.1

\begin{equation}
\int \mathrm{sn}^{2} = \int \frac{1}{k^2}\left[ 1 – \mathrm{dn}^{2}\right] = \frac{1}{k^2}\left[u -\mathrm{E}\right]
\end{equation}
We used GR 5.134.3.

GR 5.134.2

\begin{align}
\int \mathrm{cn}^{2} &= \int [1-\mathrm{sn}^{2}] = u-\frac{u}{k^2}+\frac{\mathrm{E}}{k^2} \\
&= \frac{1}{k^2}\left[\mathrm{E} \,-{k^{\prime}}^{2}u \right]
\end{align}
We used GR 5.134.3.

Continue reading “Integrals of Gradshteyn and Ryzhik: 5.134 – Integrals of Squares of Jacobi Elliptic Functions – Revised”

Derivatives of Jacobi Elliptic Functions – Part 2

Here we derive derivatives of combinations of Jacobi elliptic functions that will be useful in the evaluation of integrals of Jacobi elliptic functions which will be presented in future posts. We make use of results from Derivatives of Jacobi Elliptic Functions and Relationships Between Squares of Jacobi Elliptic Functions.

\begin{equation}
\frac{\partial}{\partial u} \mathrm{cn}^2 = -2\mathrm{cn}\,\mathrm{dn}\,\mathrm{sn}
\label{eq:dpjef-1}
\tag{1}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{dn}^2 = -2k^{2}\mathrm{cn}\,\mathrm{dn}\,\mathrm{sn}
\label{eq:dpjef-2}
\tag{2}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{sn}^2 = 2\mathrm{cn}\,\mathrm{dn}\,\mathrm{sn}
\label{eq:dpjef-3}
\tag{3}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{nc}^2 = \frac{\partial}{\partial u} \frac{1}{\mathrm{cn}^2}
= 2\,\mathrm{nc}\,\mathrm{sc}\,\mathrm{dc}
= 2\frac{\mathrm{dn}\,\mathrm{sn}}{\mathrm{cn}^3}
\label{eq:dpjef-4}
\tag{4}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{nd}^2 = \frac{\partial}{\partial u} \frac{1}{\mathrm{dn}^2}
= 2k^{2}\mathrm{nd}\,\mathrm{sd}\,\mathrm{cd}
= 2k^{2}\frac{\mathrm{cn}\,\mathrm{sn}}{\mathrm{dn}^3}
\label{eq:dpjef-5}
\tag{5}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{ns}^2 = \frac{\partial}{\partial u} \frac{1}{\mathrm{sn}^2}
= -2\,\mathrm{ns}\,\mathrm{cs}\,\mathrm{ds}
= -2\frac{\mathrm{cn}\,\mathrm{dn}}{\mathrm{sn}^3}
\label{eq:dpjef-6}
\tag{6}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{cn}\,\mathrm{dn} = -k^{2}\mathrm{cn}^{2}\mathrm{sn}\,-\mathrm{dn}^{2}\mathrm{sn}
\label{eq:dpjef-7}
\tag{7}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{cn}\,\mathrm{sn} = \mathrm{cn}^{2}\,\mathrm{dn} –
\mathrm{dn}\,\mathrm{sn}^{2}
\label{eq:dpjef-8}
\tag{8}
\end{equation}

\begin{equation}
\frac{\partial}{\partial u} \mathrm{dn}\,\mathrm{sn} = 2\mathrm{cn}\,\mathrm{dn}^{2}\,-\mathrm{cn}
\label{eq:dpjef-9}
\tag{9}
\end{equation}

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.

Integrals of Gradshteyn and Ryzhik: 5.136 – 5.139 – Integrals of Combinations of Jacobi Elliptic Functions (Part 1)

5.136.1

\begin{equation}
\int \mathrm{sn}\,\mathrm{cn} = -\frac{1}{k^2}\mathrm{dn}
\end{equation}

5.136.2

\begin{equation}
\int \mathrm{sn}\,\mathrm{dn} = -\mathrm{cn}
\end{equation}

5.136.3

\begin{equation}
\int \mathrm{cn}\,\mathrm{dn} = \mathrm{sn}
\end{equation}
The three integrals above follow from the derivatives of the Jacobi elliptic functions.

5.137.1

\begin{equation}
\int \frac{\mathrm{sn}}{\mathrm{cn}^2} = \frac{1}{{k^{\prime}}^2} \frac{\mathrm{dn}}{\mathrm{cn}} = \frac{1}{{k^{\prime}}^2} \mathrm{dc}
\end{equation}

5.137.2

\begin{equation}
\int \frac{\mathrm{sn}}{\mathrm{dn}^2} = -\frac{1}{{k^{\prime}}^2} \frac{\mathrm{cn}}{\mathrm{dn}}
= -\frac{1}{{k^{\prime}}^2} \mathrm{cd}
\end{equation}

5.137.3

\begin{equation}
\int \frac{\mathrm{cn}}{\mathrm{sn}^2} = -\frac{\mathrm{dn}}{\mathrm{sn}} = -\mathrm{ds}
\end{equation}

5.137.4

\begin{equation}
\int \frac{\mathrm{cn}}{\mathrm{dn}^2} = \frac{\mathrm{sn}}{\mathrm{dn}} = \mathrm{sd}
\end{equation}

5.137.5

\begin{equation}
\int \frac{\mathrm{dn}}{\mathrm{sn}^2} = -\frac{\mathrm{cn}}{\mathrm{sn}} = -\mathrm{cs}
\end{equation}

5.137.6

\begin{equation}
\int \frac{\mathrm{dn}}{\mathrm{cn}^2} = \frac{\mathrm{sn}}{\mathrm{cn}} = \mathrm{sc}
\end{equation}
The six integrals above follow from the derivatives of the Jacobi elliptic functions.

5.138.1

\begin{align}
\int \frac{\mathrm{cn}}{\mathrm{sn}\,\mathrm{dn}} &= \int \frac{\mathrm{cn}}{\mathrm{sn}\,\mathrm{dn}} \frac{\mathrm{dn}}{\mathrm{dn}} = \int \frac{\mathrm{cn}}{\mathrm{dn}^2} \frac{1}{\mathrm{sd}} \\
&= \int \frac{1}{w} = \ln(\mathrm{sd}) = \ln\left(\frac{\mathrm{sn}}{\mathrm{dn}}\right)
\end{align}
We used the substitution \(w=\mathrm{sd}\).

5.138.2

\begin{align}
\int \frac{\mathrm{sn}}{\mathrm{cn}\,\mathrm{dn}} &= \int \frac{\mathrm{sn}}{\mathrm{cn}\,\mathrm{dn}} \frac{\mathrm{cn}}{\mathrm{cn}} = \int \frac{\mathrm{sn}}{\mathrm{cn}^2} \frac{1}{\mathrm{dc}} \\
&= \frac{1}{{k^{\prime}}^2} \int \frac{1}{w} = \frac{1}{{k^{\prime}}^2} \ln(\mathrm{dc}) = \frac{1}{{k^{\prime}}^2} \ln\left(\frac{\mathrm{dn}}{\mathrm{cn}}\right)
\end{align}
We used the substitution \(w=\mathrm{dc}\).

5.138.3

\begin{align}
\int \frac{\mathrm{dn}}{\mathrm{sn}\,\mathrm{cn}} &= \int \frac{\mathrm{dn}}{\mathrm{sn}\,\mathrm{cn}} \frac{\mathrm{cn}}{\mathrm{cn}} = \int \frac{\mathrm{dn}}{\mathrm{cn}^2} \frac{1}{\mathrm{sc}} \\
&= \int \frac{1}{w} = \ln(\mathrm{sc}) = \ln\left(\frac{\mathrm{sn}}{\mathrm{cn}}\right)
\end{align}
We used the substitution \(w=\mathrm{sc}\).

5.139.1

\begin{equation}
\int \frac{\mathrm{cn}\,\mathrm{dn}}{\mathrm{sn}} = \int \frac{1}{w} = \ln(\mathrm{sn})
\end{equation}

5.139.2

\begin{align}
\int \frac{\mathrm{sn}\,\mathrm{dn}}{\mathrm{cn}} &= \int \frac{\mathrm{sn}\,\mathrm{dn}}{\mathrm{cn}} \frac{\mathrm{cn}}{\mathrm{cn}} = \int \frac{\mathrm{sn}\,\mathrm{dn}}{\mathrm{cn}^2} \frac{1}{\mathrm{nc}} \\
&= \int \frac{1}{w} = \ln(\mathrm{nc}) = \ln\left(\frac{1}{\mathrm{cn}}\right)
\end{align}
We used the substitution \(w=\mathrm{nc}\).

5.139.3

\begin{equation}
\int \frac{\mathrm{cn}\,\mathrm{sn}}{\mathrm{dn}} = -\frac{1}{k^2} \int \frac{1}{w} = -\frac{1}{k^2} \ln(\mathrm{dn})
\end{equation}

Derivatives of Jacobi Elliptic Functions

Here we compute derivatives of 9 of the Jacobi elliptic functions with respect to the argument \(u\). Three were derived here, and are reproduced below.

For convenience, we drop the argument and modulus.

\begin{equation}
\frac{\partial \,\mathrm{sn}}{\partial u} = \mathrm{cn}\,\mathrm{dn}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{cn}}{\partial u} = -\mathrm{dn}\,\mathrm{sn}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{dn}}{\partial u} = -k^{2}\mathrm{cn}\,\mathrm{sn}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{ns}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{sn}} = -\frac{\mathrm{cn}\,\mathrm{dn}}{\mathrm{sn}^2} = -\mathrm{cs}\,\mathrm{ds}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{nc}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{cn}} = \frac{\mathrm{dn}\,\mathrm{sn}}{\mathrm{cn}^2} = \mathrm{sc}\,\mathrm{dc}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{nd}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{dn}} = k^{2}\frac{\mathrm{cn}\,\mathrm{sn}}{\mathrm{dn}^2} = k^{2}\mathrm{sd}\,\mathrm{cd}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{cs}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{sc}}
= \frac{\partial}{\partial u}\frac{\mathrm{cn}}{\mathrm{sn}} = -\frac{\mathrm{dn}}{\mathrm{sn}^2}
= -\mathrm{dn}\,\mathrm{ns}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{sd}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{ds}}
= \frac{\partial}{\partial u}\frac{\mathrm{sn}}{\mathrm{dn}} = \frac{\mathrm{cn}}{\mathrm{dn}^2}
= \mathrm{cn}\,\mathrm{nd}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{ds}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{sd}}
= \frac{\partial}{\partial u}\frac{\mathrm{dn}}{\mathrm{sn}} = -\frac{\mathrm{cn}}{\mathrm{sn}^2}
= -\mathrm{cn}\,\mathrm{ns}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{sc}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{cs}}
= \frac{\partial}{\partial u}\frac{\mathrm{sn}}{\mathrm{cn}} = \frac{\mathrm{dn}}{\mathrm{cn}^2}
= \mathrm{nc}\,\mathrm{dc}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{cd}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{dc}}
= \frac{\partial}{\partial u}\frac{\mathrm{cn}}{\mathrm{dn}} = -{k^{\prime}}^2\frac{\mathrm{sn}}{\mathrm{dn}^2}
= -{k^{\prime}}^2\mathrm{sn}\,\mathrm{nd}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{dc}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{cd}}
= \frac{\partial}{\partial u}\frac{\mathrm{dn}}{\mathrm{cn}} = {k^{\prime}}^2\frac{\mathrm{sn}}{\mathrm{cn}^2}
= {k^{\prime}}^2\mathrm{sn}\,\mathrm{nc}^2
\end{equation}