Summing series through Fourier series

In summary, to sum the Hermite series S(a)=1+exp(-a^2)+exp(-4a^2)+exp(-9a^2)+..., we can use the Fourier transform and its inverse to find a relation between S(a) and S(pi/a). Alternatively, we can use software or online tools to calculate the Fourier transform and inverse for us. The Fourier transform can be expressed as an integral and can be a complex and time-consuming process to calculate by hand.
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Hi, I was trying to sum the Series S(a)=1+exp(-a^2)+exp(-4a^2)+exp(-9a^2)+... According to the notes where I found it it could be done through Fourier Series. I managed to find a relation between S(a) and S(pi/a), and it works, but I can't find S(a) alone. Can anybody help me find a way to do it? Thank you in advance
 
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Hello there,

I am a scientist and I would be happy to help you with this problem. The series you are trying to sum is known as the Hermite series, and it is commonly used in physics and mathematics to describe certain physical phenomena. In order to find a way to sum this series, we can use the Fourier transform.

The Fourier transform is a mathematical tool that can be used to decompose a function into a series of sinusoidal functions. In this case, we can use it to decompose the function S(a) into a series of sinusoidal functions. This will allow us to find a relation between S(a) and S(pi/a), which you have already found.

To find S(a) alone, we can use the inverse Fourier transform. This will allow us to reconstruct the original function S(a) from its Fourier transform. However, in order to do this, we need to know the Fourier transform of S(a).

The Fourier transform of S(a) can be expressed as:

S(a) = ∫ -∞ to ∞ f(x) e^(-2πiax)dx

where f(x) is the function that we want to transform. In your case, f(x) = 1 + exp(-a^2) + exp(-4a^2) + exp(-9a^2) + ...

Using this formula, we can calculate the Fourier transform of S(a) and then use the inverse Fourier transform to find S(a) alone. However, this can be a complex and time-consuming process.

Alternatively, we can use a computer program or software to calculate the Fourier transform and the inverse Fourier transform for us. This will save us time and effort. There are many programs available that can do this, such as MATLAB, Mathematica, or even online tools like Wolfram Alpha.

I hope this helps you in finding a way to sum the Hermite series. Let me know if you have any further questions or need any additional assistance. Best of luck!
 

What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of trigonometric functions. It can be used to approximate a wide range of functions and is commonly used in signal processing, engineering, and physics.

How are Fourier series used to sum series?

Fourier series can be used to sum series by representing the series as a periodic function and then approximating it with a Fourier series. The more terms included in the series, the closer the approximation will be to the actual sum.

What is the formula for a Fourier series?

The formula for a Fourier series is given by:
f(x) = a0/2 + ∑[ancos(nx) + bnsin(nx)],
where a0, an, and bn are the coefficients of the series and n is the number of terms in the series.

What is the difference between a Fourier series and a Taylor series?

A Taylor series approximates a function at a specific point, while a Fourier series approximates a periodic function over its entire domain. Additionally, a Fourier series uses trigonometric functions while a Taylor series uses polynomial functions.

How are Fourier series used in real-world applications?

Fourier series are used in a variety of real-world applications, such as signal processing, image compression, and audio compression. They are also used in physics and engineering to model periodic phenomena. Additionally, Fourier series have applications in mathematics, such as in solving partial differential equations.

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