Seeking for help to find a formula for $\int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}}$, where $a>1.$

Question

When tackling the question, I found that for any $a>1$,

$$ I_1(a)=\int_{0}^{\pi} \frac{d x}{a-\cos x}=\frac{\pi}{\sqrt{a^{2}-1}}. $$ Then I started to think whether there is a formula for the integral

$$ I_n(a)=\int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}}, $$ where $n\in N.$

After trying some substitution and integration by parts, I still failed and got no idea for reducing the power n. After two days, the Leibniz Rule for high derivatives come to my mind.

Differentiating $I_1(a)$ w.r.t. $a$ by $(n-1)$ times yields

$$ \displaystyle \begin{array}{l} \displaystyle \int_{0}^{\pi} \frac{(-1)^{n-1}(n-1) !}{(a-\cos x)^{n}} d x=\frac{d^{n-1}}{d a^{n-1}}\left(\frac{\pi}{\sqrt{a^{2}-1}}\right) \\ \displaystyle \int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}}=\frac{(-1)^{n-1} \pi}{(n-1) !} \frac{d^{n-1}}{d a^{n-1}}\left(\frac{1}{\sqrt{a^{2}-1}}\right) \tag{*}\label{star} \end{array} $$

I am glad to see that the integration problem turn to be merely a differentiation problem.

Now I am going to find the $(n-1)^{th} $ derivative by Leibniz Rule.

First of all, differentiating $I_1(a)$ w.r.t. $a$ yields $$ \left(a^{2}-1\right) \frac{d y}{d a}+a y=0 \tag{1}\label{diffeq} $$ Differentiating \eqref{diffeq} w.r.t. $a$ by $(n-1)$ times gets $$ \begin{array}{l} \displaystyle \left(a^{2}-1\right) \frac{d^{n} y}{d a^{n}}+\left(\begin{array}{c} n-1 \\ 1 \end{array}\right)(2 a) \frac{d^{n-1} y}{d a^{n-1}}+2\left(\begin{array}{c} n-1 \\ 2 \end{array}\right) \frac{d^{n-2} y}{d a^{n-2}}+x \frac{d^{n-1} y}{d a^{n-1}}+(n-1) \frac{d^{n-2} y}{d a^{n-2}}=0 \end{array} $$ Simplifying, $$ \left(a^{2}-1\right) y^{(n)}+(2 n-1) ay^{(n-1)}+(n-1)^{2} y^{(n-2)}=0 \tag{2}\label{diffrec} $$

Initially, we have $ \displaystyle y^{(0)}=\frac{1}{\sqrt{a^{2}-1}}$ and $ \displaystyle y^{(1)}=-\frac{a}{\left(a^{2}-1\right)^{\frac{3}{2}}}.$

By \eqref{diffrec}, we get $$ y^{(2)}=\frac{2 a^{2}+1}{\left(a^{2}-1\right)^{\frac{5}{2}}} $$ and $$ \displaystyle y^{(3)}=-\frac{3 a\left(2 a^{2}+3\right)}{\left(a^{2}-1\right)^{\frac{7}{2}}} $$ Plugging into \eqref{star} yields $$ \begin{aligned} \int_{0}^{\pi} \frac{d x}{(a-\cos x)^{3}} &=\frac{\pi}{2} y^{(2)}=\frac{\pi\left(2 a^{2}+1\right)}{2\left(a^{2}-1\right)^{\frac{5}{2}}} \\ \int_{0}^{\pi} \frac{d x}{(a-\cos x)^{4}} &=-\frac{\pi}{6} \cdot \frac{3 a\left(2 a^{2}+3\right)}{\left(a^{2}-1\right)^{\frac{7}{2}}} =-\frac{\pi a\left(2 a^{2}+3\right)}{2\left(a^{2}-1\right)^{\frac{7}{2}}} \end{aligned} $$ Theoretically, we can proceed to find $I_n(a)$ for any $n\in N$ by the recurrence relation in $(2)$ .

By Mathematical Induction, we can further prove that the formula is $$ \int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}}=\frac{\pi P(a)}{\left(a^{2}-1\right)^{\frac{2 n-1}{2}}} $$ for some polynomial $P(a)$ of degree $n-1$.

Last but not least, how to find the formula for $P(a)$? Would you help me?

Answer 1

One way using complex analysis:

Since the integrand is even: $$ I=\int_{0}^{\pi} \frac{d x}{(a-\cos x)^n} = \frac{1}{2} \int_{-\pi}^{\pi} \frac{d x}{(a-\cos x)^n} $$

Since $a>1$, we can expand the integrand with the generalized binomial theorem:

$$ I = \frac{1}{2} \int_{-\pi}^{\pi} \frac{d x}{(a-\cos x)^n} = \frac{1}{2} \int_{-\pi}^{\pi} \sum_{j=0}^{\infty} \binom{-n}{j} a^{-n-j}(-1)^j\cos^j(x) dx $$

From Fubini-Tonelli we can interchange integral and series:

$$ I = \frac{1}{2} \sum_{j=0}^{\infty} \binom{-n}{j}a^{-n-j}(-1)^j\int_{-\pi}^{\pi} \cos^j(x) dx $$

Do the following substitution

$$ \cos x = \frac{z+z^{-1}}{2}$$

$$ dx = \frac{dz}{zi}$$

The integral is now a contour integral round the unit complex circle:

$$ I = \frac{1}{2} \sum_{j=0}^{\infty} \binom{-n}{j}\frac{a^{-n-j}(-1)^j}{2^j}\oint_{|z|=1} \frac{(z^2+1)^j}{z^{j+1} }dz$$

The integral has a pole at $z=0$. To find the residue, expand the integrand:

$$\frac{(z^2+1)^j}{z^{j+1} }= \sum_{k=0}^{j}\binom{j}{k} z^{2k-j-1}$$

The residue is the coefficient of $z^{2k-j-1}= z^{-1}$

Then

$$ 2k-j-1 = -1 \Longrightarrow k = \frac{j}{2}$$

Therefore, the residue exists if $j$ is $\textbf{divisible by}$ $2$:

$$\oint_{|z|=1} \frac{(z^2+1)^j}{z^{j+1} }dx = 2\pi i \operatorname{Res}\left(\frac{(z^2+1)^j}{z^{j+1}},0\right) = 2\pi i \binom{j}{\frac{j}{2}} $$

Hence, we have

$$ I = \pi\sum_{j=0}^{\infty} \binom{-n}{2j}\binom{2j}{j}\frac{a^{-n-2j}}{2^{2j}} $$

Note

$$\binom{-n}{2j} = \frac{(-n-2j+1)_{2j}}{(2j!}$$

where $(x)_{n} = x(x+1)\cdots(x+n-1)$ is the rising factorial (Pochhammer polynomial)

and

$$\binom{2j}{j} = \frac{(2j)!}{j!^2}$$

Hence

$$\binom{-n}{2j}\binom{2j}{j} = \frac{(-n-2j+1)_{2j}}{j!^2}$$

From the reflection formula for the Pochhammer polynomial:

$$(-x)_{m} = (-1)^m(x-m+1)_{m}$$

we have $$(-n-2j+1)_{2j}= (n)_{2j}$$

and from the duplication formula for the degree of the Pochhammer polynomial:

$$(x)_{2m} = 4^m\left(\frac{x}{2}\right)\left(\frac{1+x}{2}\right)$$

we have

$$(n)_{2j} = 4^{j}\left(\frac{n}{2}\right)\left(\frac{n+1}{2}\right)$$

Hence

$$ I = \pi\sum_{j=0}^{\infty} \binom{-n}{2j}\binom{2j}{j}\frac{a^{-n-2j}}{2^{2j}} = \frac{\pi}{a^n}\sum_{j=0}^{\infty}\frac{\left(\frac{n}{2}\right)\left(\frac{n+1}{2}\right)}{(1)_{j}} \frac{\left(\frac{1}{a^2}\right)^j}{j!}$$

$$\Longrightarrow I = \frac{\pi}{a^n}{}_{2}F_{1}\left({\frac{n}{2},\frac{n+1}{2}\atop 1};\frac{1}{a^2}\right)$$

where ${}_{2}F_{1}({a,b\atop c};z)$ is the Guassian function

Now using the following forumula:

$${}_{2}F_{1}\left({a,a+\tfrac{1}{2}\atop c};z\right)=2^{c-1}z^{\frac{(1-c)}{2}}% (1-z)^{-a+\left(\frac{c-1}{2}\right)}P^{1-c}_{2a-c}\left(\frac{1}{\sqrt{1-z}}\right) $$

where $P_{n}^{(\alpha,\beta)}$ is the Jacobi polynomial

we have

$$ I =\frac{\pi}{a^n}{}_{2}F_{1}\left({\frac{n}{2},\frac{n+1}{2}\atop 1};\frac{1}{a^2}\right)= \frac{\pi}{a^n}\left(1-\frac{1}{a^2}\right)^{-\frac{n}{2}}P_{n-1}\left(\frac{a}{\sqrt{a^2-1}}\right)$$

where $P_{n}$ is the standard Legendre polynomial which satisfies the Rodrigues' formula:

$$ P_{n}(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n$$

Finally, we can conclude

$$\boxed{\int_{0}^{\pi} \frac{d x}{(a-\cos x)^n} = \frac{\pi}{2^{n-1}(n-1)!a^n}\left(1-\frac{1}{a^2}\right)^{-\frac{n}{2}}\left( \frac{d^{n-1}}{dx^{n-1}} (x^2-1)^{n-1} \right)_{x = \frac{a}{\sqrt{a^2-1}}}}$$

This is similar to the formula you found. However, Legendre polynomials also have an explicit formula:

$$P_{n} (x) = \sum_{j} (-1)^{\frac{j}{2}} \frac{(2n-j-1)!!}{j!!(n-j)!} x^{n-j} \quad j=0,2,4,...,n-\frac{1}{2}\pm \frac{1}{2}$$

(ends with $n$ in the case $n$ even and ends with $n-1$ in the case $n$ odd).

Hence

$$ \boxed{I =\int_{0}^{\pi} \frac{d x}{(a-\cos x)^n} = \frac{\pi}{a^n}\left(1-\frac{1}{a^2}\right)^{-\frac{n}{2}}\sum_{j} (-1)^{\frac{j}{2}} \frac{(2n-j-3)!!}{j!!(n-1-j)!} \left(\frac{a}{\sqrt{a^2-1}}\right)^{n-1-j} \quad j=0,2,4,...,n-\frac{3}{2}\pm \frac{1}{2} \textrm{ ($n-1$ even or odd)}} $$

It turned out that the polynomial you are looking for is this Legendre polynomial.

Answer 2

From what you have deduced, we can apply here the Faà di Bruno's formula $$ \begin{align} I_n(a) &= \frac{(-1)^{n-1}\pi}{(n-1)!} \frac{d^{n-1}}{da^{n-1}} \left(\frac{1}{\sqrt{a^2 - 1}}\right) \\ &= \frac{(-1)^{n-1}\pi}{(n-1)!} \frac{d^{n-1} \sqrt{b(a)}}{da^{n-1}} \\ &= \sum_{}\frac{(n-1)!}{\prod_{k=1}^{n-1}m_k! k!^{m_k}} \frac{d^{m_1 + \cdots + m_{n-1}}\sqrt{b}}{db^{m_1 + \cdots + m_{n-1}}} \cdot \prod_{j=1}^{n-1} \left(\frac{d^j b(a)}{da^j}\right)^{m_j} \end{align} $$ where $b = \frac{1}{a^2 - 1}$ and the summation is over all $n-1$ tuples of non-negative integers $m_i$ such that $$ \sum_{k=1}^{n-1} k m_k = n-1. $$ Indeed, we have that $\frac{db}{da} = \frac{-2a}{(a^2 - 1)^2}$ (finding a general formula for this should be not too difficult) and that for any $k\in\mathbb{N}$ $$ \frac{d^k}{db^k}\sqrt{b} = (-1)^{k-1}\frac{(2(k-1))!}{(k-1)!}(4b)^{\frac{1 - 2k}{2}}, $$ where the last equality was obtained via this question.

Following down this path and executing the necessary derivatives should yield a sufficient answer.

Answer 3

Express the integral as \begin{align} \int_{0}^{\pi} \frac{d x}{(a-\cos x)^{n}} =\sum_{k=0}^{[\frac{n-1}2]} \frac{\binom{n-1}{2k}a^{n-2k-1}}{(a^2-1)^{n-1/2}}\int_0^{\pi}\cos^{2k}x \ dx = \frac{\pi P(a)}{\left(a^{2}-1\right)^{n-{1}/{2}}} \end{align} which leads to $$P(a) = \sum_{k=0}^{[\frac{n-1}2]} \binom{n-1}{2k} \frac{(2k-1)!!}{(2k)!!} a^{n-2k-1} $$