I want to evaluate the following integral via complex analysis \int_{x=0}^{x=\infty}e^{-ax}\cos(bx)dx,\

Francisca Rodden

Francisca Rodden

Answered question

2022-01-04

I want to evaluate the following integral via complex analysis
x=0x=eaxcos(bx)dx, a>0
Which function/ contour should I consider ?

Answer & Explanation

Nadine Salcido

Nadine Salcido

Beginner2022-01-05Added 34 answers

0eaxcos(bx)dx=R(0+eaxeibxdx)
(where R(z) denotes the real part of (z). Then,
0+eaxeibxdx=0+e(aib)xdx
=limM+0M0+e(aib)xdx
=limM+[1aibe(aib)x]0M
=limM+(1aibe(aib)M+1aib)
=1aib
=a+ib|aib|2
So,
0+eaxcos(bx)dx=a|aib|2
poleglit3

poleglit3

Beginner2022-01-06Added 32 answers

Let us integrate the function eAz, where A=a2+b2 on a circular sector in the first quadrant, centered at the origin and of radius R, with angle ω which satisfies cosω=aA, and therefore sinω=bA. Let this sector be called γ.
Since our integrand is obviously holomorphic on the whole plane we get:
γdzeAz=0
Breaking it into its three pieces we obtain:
aRdxeAx+0ωdϕiReAReiϕ+R0dreiωeAreω=0
1A=1A0dr(a+ib)er(a+ib)
0dr(a+ib)ear(cosbrisinbr)=1
Now let's call Ic=0drearcosbr and Is=0drearsinbr, then:
aIciaIs+ibIc+bIs=1
and by solving:
aIc+bIs=1; aIs+bIc=0
Ic=aa2+b2; Is=ba2+b2
This method relies only on the resource of contour integration as you asked!
star233

star233

Skilled2022-01-11Added 403 answers

First,you may use the definition of Lapace transform to get it or integration by parts twice.
For complex numbers your integrand is the real part of exp(ax+ibx). Use this function to evaluate the unbounded integral then evaluate the bounded one

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