• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Fourier series for a series of functions

Problem Statement
From equations 1) or 2) use the appropriate values of ## \theta ## usually ## (0 ,\frac{\pi }{2},\pi )## , to get to the indicated equation.
Relevant Equations
1)

## f(\theta )=e^{b\theta }(-\pi <\theta <\pi ) \quad | \quad\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ##


2)

##f(\theta )=e^{b\theta }(0<\theta <2\pi )\quad | \quad\frac{e^{2\pi b}-1}{2\pi }\sum_{-\infty }^{\infty }\frac{e^{in\theta }}{b-in}##


Indicated equation:

##\sum_{1 }^{\infty }\frac{(-1)^{n}}{n^{2}+b^{2}}=\frac{\pi }{2b}csch(b\pi )-\frac{1}{2b^{2}}##
## ##

Well I start with equation 1):


## e^{b\theta }=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ##

If ## \theta =0 ##



##e^{b(0)}=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in(0) }##



##1=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in} ##



Using the conjugate of the complex:



##1=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}\frac{b+in}{b+in} ##



## 1=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }(-1)^{n}\frac{b+in}{b^{2}+n^{2}}##



## \pi csch(b\pi )=\sum_{-\infty }^{\infty }(-1)^{n}\frac{b+in}{b^{2}+n^{2}}##


Up to now I have only arrived, I do not know if it is for equation 1 to arrive at the result or for equation 2, another doubt that I have is that the summations of equations 1) and 2) tend to ##(-\infty ,\infty ) ## and the result of the summation tends to ## (1,\infty )## , as I do so that the sum of ## (-\infty ,\infty )## tend a ## (1,\infty )## ,my idea was to see if the functions were even or odd but since both are exponential they are not odd or even, they can help me with my problem in advance thanks.
 

Attachments

pasmith

Homework Helper
1,668
369
The terms for [itex]n > 0[/itex] and [itex]-n[/itex] are complex conjugates. This reordering of terms is justifiable if you treat the summation as [tex]
\lim_{N \to \infty} \sum_{n = -N}^{N} (-1)^n \frac{b + in}{b^2 + n^2}.[/tex]
 

Want to reply to this thread?

"Fourier series for a series of functions" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top