calculate-A-m-0-sin-mx-e-2pix-1-dx-with-m-gt-0-

Question Number 53600 by maxmathsup by imad last updated on 23/Jan/19

c a l c u l a t e A_{m} = \int_{0}^{\infty} \frac{s i n (m x)}{e^{2 π x} - 1} d x w i t h m > 0

Commented by maxmathsup by imad last updated on 24/Jan/19

w e h a v e A_{m} = \int_{0}^{\infty} \frac{e^{- 2 π x} s i n (m x)}{1 - e^{- 2 π x}} d x = I m (\int_{0}^{\infty} \frac{e^{- 2 π x} e^{i m x}}{1 - e^{- 2 π x}} d x) b u t

\int_{0}^{\infty} \frac{e^{- (2 π - i m) x}}{1 - e^{- 2 π x}} d x = \int_{0}^{\infty} e^{- (2 π - i m) x} (\sum_{p = 0}^{\infty} e^{- 2 π p x})

Missing \left or extra \right

= \sum_{p = 0}^{\infty} \frac{1}{(2 + 2 p) π - i m} = \sum_{p = 0}^{\infty} \frac{(2 + 2 p) π + i m}{{(2 + 2 p)}^{2} π^{2} + m^{2}} \Rightarrow

A_{m} = \sum_{p = 0}^{\infty} \frac{m}{4 {(p + 1)}^{2} π^{2} + m^{2}} a n d A_{m} c a n b e c a l c u l a t e d b y f o u r i e r s e r i e s

\dots . b e c o n t i n u e d \dots