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: 2.671 and 2.672 – Combinations of trigonometric and hyperbolic functions

Here, integration by parts is used twice for each integral. We repeatedly use the following:
\begin{equation}
u=\sinh(ax+b) \quad du=a\,\cosh(ax+b) \quad \mathrm{and} \quad u=\cosh(ax+b) \quad du=a\,\sinh(ax+b)
\end{equation}
\begin{equation}
dv=\sin(cx+d) \quad v=-\frac{\cos(cx+d)}{c} \quad \mathrm{and} \quad dv=\cos(cx+d) \quad v=\frac{\sin(cx+d)}{c}
\end{equation}

2.671.1

\begin{equation}
\int\sinh(ax+b)\sin(cx+d) dx = -\frac{\sinh(ax+b)\cos(cx+d)}{c} + \frac{a}{c}\int\cosh(ax+b)\cos(cx+d) dx
\end{equation}
\begin{equation}
= -\frac{\sinh(ax+b)\cos(cx+d)}{c} + \frac{a}{c} \left[ \frac{\cosh(ax+b)\sin(cx+d)}{c}\,-\frac{a}{c}\int\sinh(ax+b)\sin(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\cosh(ax+b)\sin(cx+d)\,-c\,\sinh(ax+b)\cos(cx+d)}{a^2 + c^2}
\end{equation}

2.671.2

\begin{equation}
\int\sinh(ax+b)\cos(cx+d) dx = \frac{\sinh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c}\int\cosh(ax+b)\sin(cx+d) dx
\end{equation}
\begin{equation}
= \frac{\sinh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c} \left[-\frac{\cosh(ax+b)\cos(cx+d)}{c}+\frac{a}{c}\int\sinh(ax+b)\cos(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\cosh(ax+b)\cos(cx+d) + c\,\sinh(ax+b)\sin(cx+d)}{a^2 + c^2}
\end{equation}

2.671.3

\begin{equation}
\int\cosh(ax+b)\sin(cx+d) dx = -\frac{\cosh(ax+b)\cos(cx+d)}{c} + \frac{a}{c}\int\sinh(ax+b)\cos(cx+d) dx
\end{equation}
\begin{equation}
= -\frac{\cosh(ax+b)\cos(cx+d)}{c} + \frac{a}{c} \left[ \frac{\sinh(ax+b)\sin(cx+d)}{c}\,-\frac{a}{c}\int\cosh(ax+b)\sin(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\sinh(ax+b)\sin(cx+d)\,-c\,\cosh(ax+b)\cos(cx+d)}{a^2 + c^2}
\end{equation}

2.671.4

\begin{equation}
\int\cosh(ax+b)\cos(cx+d) dx = \frac{\cosh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c}\int\sinh(ax+b)\sin(cx+d) dx
\end{equation}
\begin{equation}
= \frac{\cosh(ax+b)\sin(cx+d)}{c} \,- \frac{a}{c} \left[-\frac{\sinh(ax+b)\cos(cx+d)}{c}+\frac{a}{c}\int\cosh(ax+b)\cos(cx+d) dx \right]
\end{equation}
\begin{equation}
= \frac{a\,\sinh(ax+b)\cos(cx+d) + c\,\cosh(ax+b)\sin(cx+d)}{a^2 + c^2}
\end{equation}

2.672.1

Let \(a=c=1,\,b=d=0\) in 2.671.1
\begin{equation}
\int\sinh(x)\sin(x) dx = \frac{1}{2}[\cosh(x)\sin(x)\,- \sinh(x)\cos(x)]
\end{equation}

2.672.2

Let \(a=c=1,\,b=d=0\) in 2.671.2
\begin{equation}
\int\sinh(x)\cos(x) dx = \frac{1}{2}[\sinh(x)\sin(x)+ \cosh(x)\cos(x)]
\end{equation}

2.672.3

Let \(a=c=1,\,b=d=0\) in 2.671.3
\begin{equation}
\int\cosh(x)\sin(x) dx = \frac{1}{2}[\sinh(x)\sin(x)\,- \cosh(x)\cos(x)]
\end{equation}

2.672.4

Let \(a=c=1,\,b=d=0\) in 2.671.4
\begin{equation}
\int\cosh(x)\cos(x) dx = \frac{1}{2}[\cosh(x)\sin(x)+ \sinh(x)\cos(x)]
\end{equation}

Integrals of Gradshteyn and Ryzhik: 2.723 – Combinations of Logarithms and Algebraic Functions

The strategy for all three integrals is to use a different integrand, parameterized, differentiate under the integral, then apply a limit to recover the desired integral.

2.723.1

The integral to be evaluated is
\begin{equation}
I = \int x^{n}\ln(x) dx
\end{equation}
Let
\begin{equation}
I(a) = \int x^{a-1} dx = \frac{x^a}{a}
\end{equation}
so that
\begin{align}
\frac{\partial}{\partial a}I(a) &= \int x^{a-1}\ln(x) dx \\
&= \frac{\partial}{\partial a} \frac{x^a}{a} = x^{a} \left[\frac{\ln(x)}{a}\,- \frac{1}{a^2} \right]
\end{align}

\begin{equation}
\lim_{a \to n+1} \frac{\partial}{\partial a}I(a) \Rightarrow
I = \bbox[5px,border:2px solid blue] {\int x^{n}\ln(x) dx = x^{n+1} \left[\frac{\ln(x)}{n+1}\,- \frac{1}{(n+1)^2} \right]}
\end{equation}

2.723.2

The integral to be evaluated is
\begin{equation}
I = \int x^{n}\ln^{2}(x) dx
\end{equation}

\begin{align}
\frac{{\partial}^2}{\partial a^2}I(a) &= \int x^{a-1}\ln^{2}(x) dx \\
&= \frac{\partial}{\partial a} \left[\ln(x) \frac{x^a}{a}\,- \frac{x^a}{a^2} \right]
= x^{a} \left[\frac{\ln^{2}(x)}{a}\, – \frac{2\ln(x)}{a^2} + \frac{2}{a^3} \right]
\end{align}
\begin{equation}
\lim_{a \to n+1} \frac{{\partial}^2}{\partial a^2}I(a) \Rightarrow
I = \bbox[5px,border:2px solid blue] {\int x^{n}\ln^{2}(x) dx
= x^{n+1} \left[\frac{\ln^{2}(x)}{n+1}\, – \frac{2\ln(x)}{(n+1)^2} + \frac{2}{(n+1)^3} \right]}
\end{equation}

2.723.3

The integral to be evaluated is
\begin{equation}
I = \int x^{n}\ln^{3}(x) dx
\end{equation}

\begin{align}
\frac{{\partial}^3}{\partial a^3}I(a) &= \int x^{a-1}\ln^{3}(x) dx \\
&= \frac{\partial}{\partial a} \left( x^{a} \left[\frac{\ln^{2}(x)}{a}\, – \frac{2\ln(x)}{a^2} + \frac{2}{a^3} \right] \right) \\
&= x^{a} \left[\frac{\ln^{3}(x)}{a} \,- \frac{3\ln^{2}(x)}{a^2} + \frac{6\ln(x)}{a^3} \,- \frac{6}{a^4} \right]
\end{align}

\begin{equation}
\lim_{a \to n+1} \frac{{\partial}^2}{\partial a^3}I(a) \Rightarrow
\end{equation}
\begin{equation}
I = \bbox[5px,border:2px solid blue] {\int x^{n}\ln^{2}(x) dx
= x^{n+1} \left[\frac{\ln^{3}(x)}{n+1} \,- \frac{3\ln^{2}(x)}{(n+1)^2} + \frac{6\ln(x)}{(n+1)^3} \,- \frac{6}{(n+1)^4} \right]}
\end{equation}