Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

Question

How Can I evaluate the following sum$$\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\lambda_0 a)}$$

I have read this in a research paper I have tried evaluating the sum using finite cosine transform We have $$\frac{2}{a}\frac{1}{\lambda_0}+\frac{2}{a}\sum_{m=0}^{\infty}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=f(x)$$ So $$\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}=\int_{0}^{a} f(x)\cos\left(\frac{m\pi x}{a}\right) dx $$

How to find $f(x)$ ?

And Is there any other way to evaluate the sum ?

Thanks In advance.

Answer 1

Too long for the comment.

The issue sum $$S_1 = \sum_{m=0}^{\infty}\frac{2-\delta_{m0}}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right),\tag1$$

where $\delta_{ij}$ is the Kronekker symbol,

can be considered as the particular case of $S_k$, where $$S_k = \frac{1}{a}\frac{1}{\lambda_0} + \frac{2}{a}\sum_{m=1}^{\infty}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})^k}\cos\left(\frac{m\pi x}{a}\right).\tag2$$

Using known exponential Fourier series for cosine function

$$\cos(bz) = \dfrac{2b\sin(b\pi)}\pi\left(\dfrac1{2b^2}-\sum\limits_{m=1}^\infty \dfrac{(-1)^m\cos(mz)}{m^2-b^2}\right)\tag3$$

with the values $$b=\dfrac{\lambda_0a}\pi,\quad z = \dfrac{\pi x}a,$$

one can get $$\dfrac 1\pi\dfrac{\cos(\lambda_0x)}{\sin(\lambda_0a)} = \dfrac1{a\lambda_0} + \dfrac{2\lambda_0a}{\pi^2} \sum\limits_{m=1}^\infty \dfrac{(-1)^m\cos\left(\dfrac{m\pi x}{a}\right)}{\dfrac{\lambda_0^2a^2}{\pi^2}-m^2}=S_2.$$

Answer 2

Focusing on the main task of the problem in the OP I shall calculate the sum starting at $m=1$ and dropping the overall factor $\lambda$, i.e. the expression

$$f(x) = \frac{2 }{a} \sum _{m=1}^{\infty } \frac{(-1)^m \cos \left(\frac{\pi m x}{a}\right)}{\lambda ^2-\frac{\pi m}{a}}\tag{1}$$

Assuming $\lambda ^2 \lt \frac{\pi}{a}$ we write

$$\frac{1}{\lambda ^2-\frac{\pi m}{a}} = - \int_{0}^{\infty} e^{-t (\frac{\pi m}{a}-\lambda ^2)}\, dt\tag{2}$$

and

$$\cos(\frac{m \pi x}{a}) = \Re \exp(i \frac{m \pi x}{a})$$

Substituting this into $(1)$ the /geometric) sum can be done under the integral:

$$\sum _{m=1}^{\infty } (-1)^m \exp \left(\frac{i \pi m x}{a}\right) \exp \left(-t \left(\frac{\pi m}{a}-\lambda ^2\right)\right)\\= -\frac{e^{t \left(-\left(\frac{\pi }{a}-\lambda ^2\right)\right)+\frac{\pi t}{a}+\frac{i \pi x}{a}}}{e^{\frac{\pi t}{a}}+e^{\frac{i \pi x}{a}}}$$

Integrating the negative this over $t$ according to $(2)$ and applying the missing factor gives

$$-\frac{2}{\pi} \left(-e^{\frac{i \pi x}{a}}\right)^{\frac{a \lambda ^2}{\pi }} B(-e^{\frac{i \pi x}{a}},1-\frac{a \lambda ^2}{\pi },0)\tag{3}$$

Here $B$ is the incomplete Beta function defined by

$$B(z,a,b) = \int_{0}^{z} t^{a-1} (1-t)^{b-1}$$

We now have to take the real part.

Selecting the simplified case $\lambda \to 0$ expression $(3)$ reduces to

$$\frac{2}{\pi} \log \left(1+e^{\frac{i \pi x}{a}}\right)$$

The real part and hence the sum is

$$f(\lambda\to 0)=\frac{1}{\pi} \log \left(4 \cos \left(\frac{\pi x}{2 a}\right)^2\right)\tag{4}$$

The case $0 \lt \lambda ^2 (\lt \frac{\pi}{a})$ can be extracted from $(3)$ as well. It will be left as an exercise in complex arithmetic to the reader.

For any $\lambda \ne \sqrt{\frac{\pi m}{a}}$ the sum can be expressed by the Hurwitz-Lerch transcendent as follows

$$f= \frac{1}{\pi} \left( e^{\frac{i \pi x}{a}} \Phi \left(-e^{\frac{i \pi x}{a}},1,1-\frac{a \lambda ^2}{\pi }\right) +e^{-\frac{i \pi x}{a}} \Phi \left(-e^{-\frac{i \pi x}{a}},1,1-\frac{a \lambda ^2}{\pi }\right) \right)\tag{5}$$

$\Phi $ is defined as

$$\Phi(z,s,a) = \sum_{k=0}^{\infty} z^k (k+a)^{-s} $$