Question

Find the Complex form of Fourier series of the function f(x) = e−ax, 0<x<π and hence deduce that sinh ax.​

Answer 1

Let the function

f

(

x

)

be defined on the interval

[

−

π

,

π

]

.

Using the well-known Euler’s formulas

cos

φ

=

e

i

φ

+

e

−

i

φ

2

,

sin

φ

=

e

i

φ

−

e

−

i

φ

2

i

,

we can write the Fourier series of the function in complex form: