# Series and product development (Ahlfors)

• gianeshwar
In summary, it appears that the Basel problem is related to p-series, and that the solution can be found by manipulating Jenson's formulae and Mittag Lefflers theorems.
gianeshwar

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Hello. It seems like these series are related to p-series, with n^2 it is more like the Basel problem (https://en.wikipedia.org/wiki/Basel_problem). I do not immediately see the proper conversion, but I will continue to think about it.

Thank you very much RUber.I was struggling with these problems after studying Jensons formulae and Mittag Lefflers theorems.

I had to ask wolframalpha.com, but was provided with the following solutions. These types of problems usually require a great deal of manipulation to show, even when the answer is known.

http://www4f.wolframalpha.com/Calculate/MSP/MSP652520diec13gdedd2dh00003i3bedh373981c66?MSPStoreType=image/gif&s=20&w=168.&h=45.
My image didn't copy properly, so here it is in Tex.
##\sum_{-\infty}^{\infty} \frac{1}{z^2 - n^2 } = \frac{\pi \cot(\pi z)}{z}##

##\sum_{-\infty}^{\infty} \frac{1}{a^2 +(z + n)^2 } = \frac{\pi \sinh(2\pi a)}{a( \cosh(2 \pi a ) - \cos(2\pi z))}##

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Thanks RUber! I have progressed a little further.Will share soon.

A bit further up on the page, marked as (11) it says: $\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^{2}-n^{2}}$. Keep that in mind. Now $\sum_{-\infty}^{\infty}\frac{1}{z^{2}-n^{2}}=\sum_{n=-\infty}^{-1}\frac{1}{z^{2}-n^{2}}+\frac{1}{z^{2}}+\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}}$. Since (-n)2=n2, we have $\sum_{-\infty}^{\infty}\frac{1}{z^{2}-n^{2}}=\frac{1}{z^{2}}+2 \cdot\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}}$. Now we are almost there - divide the expression in (11) by z (remembering to keep away from 0), you have $\frac{\pi\cot(\pi z)}{z}=\frac{1}{z^{2}}+2\cdot\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}}$. Conclusion: $\sum_{-\infty}^{\infty}\frac{1}{z^{2}-n^{2}}=\frac{\pi\cot(\pi z)}{z}$.

Thanks svein!

I shall give you some hints:
1. $\frac{1}{(z+n)^{2}+a^{2}}=\frac{1}{2ia}(\frac{1}{z+n-ia}-\frac{1}{z+n-ia})$
2. $\sum_{n=-\infty}^{\infty}\frac{1}{z+n+ia}=\sum_{n=-\infty}^{\infty}\frac{1}{z-n+ia}$
3. Let w = z+ia. We already know that $\sum_{n=-\infty}^{\infty}\frac{1}{w-n}=\pi\cot\pi w$.
4. Do the same thing with w = z-ia.
5. Substituting back, you get $\sum_{-\infty}^{\infty}\frac{1}{(z+n)^{2}+a^{2}}=\frac{1}{2ia}(\pi\cot\pi (z+ia)-\pi\cot\pi (z-ia))$
The rest is left to the student...

gianeshwar
Thank You Svein!

## 1. What is series and product development in mathematics?

Series and product development, also known as infinite series and infinite product, is a mathematical concept that deals with the sum and product of an infinite sequence of terms. It is a fundamental concept in calculus and is often used to approximate functions or solve equations.

## 2. What is the difference between a series and a product?

The main difference between a series and a product is the operation used. A series is the sum of an infinite sequence of terms, while a product is the multiplication of an infinite sequence of terms. In series, the terms are added together, while in products, the terms are multiplied together.

## 3. How is a series or product evaluated?

A series or product is evaluated by finding its limit as the number of terms approaches infinity. This can be done using various techniques, such as the ratio test, the integral test, or the comparison test. In some cases, a closed form solution can be found.

## 4. What are some real-life applications of series and product development?

Series and product development have numerous applications in real life, such as in physics, engineering, and finance. They are used to model and solve problems related to continuous growth and decay, such as population growth, radioactive decay, and compound interest.

## 5. Are there any limitations to series and product development?

Yes, there are limitations to series and product development. In some cases, the series or product may not converge, meaning it does not have a finite sum or product. Additionally, some functions may not be expressible as a series or product, making it difficult to use these techniques to solve equations involving those functions.

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