Francisca Rodden

2022-01-04

I want to evaluate the following integral via complex analysis

${\int}_{x=0}^{x=\mathrm{\infty}}{e}^{-ax}\mathrm{cos}\left(bx\right)dx,\text{}a0$

Which function/ contour should I consider ?

Which function/ contour should I consider ?

Nadine Salcido

Beginner2022-01-05Added 34 answers

(where

So,

poleglit3

Beginner2022-01-06Added 32 answers

Let us integrate the function $e}^{-Az$ , where $A=\sqrt{{a}^{2}+{b}^{2}}$ on a circular sector in the first quadrant, centered at the origin and of radius R, with angle $\omega$ which satisfies $\mathrm{cos}\omega =\frac{a}{A}$ , and therefore $\mathrm{sin}\omega =\frac{b}{A}$ . Let this sector be called $\gamma$ .

Since our integrand is obviously holomorphic on the whole plane we get:

${\oint}_{\gamma}dz{e}^{-Az}=0$

Breaking it into its three pieces we obtain:

${\int}_{a}^{R}dx{e}^{-Ax}+{\int}_{0}^{\omega}d\varphi iR{e}^{-AR{e}^{i\varphi}}+{\int}_{R}^{0}dr{e}^{i\omega}{e}^{-Ar{e}^{\omega}}=0$

$\frac{1}{A}=\frac{1}{A}{\int}_{0}^{\mathrm{\infty}}dr(a+ib){e}^{-r(a+ib)}$

${\int}_{0}^{\mathrm{\infty}}dr(a+ib){e}^{-ar}(\mathrm{cos}br-i\mathrm{sin}br)=1$

Now let's call${I}_{c}={\int}_{0}^{\mathrm{\infty}}dr{e}^{-ar}\mathrm{cos}br$ and ${I}_{s}={\int}_{0}^{\mathrm{\infty}}dr{e}^{-ar}\mathrm{sin}br$ , then:

$a{I}_{c}-ia{I}_{s}+ib{I}_{c}+b{I}_{s}=1$

and by solving:

$a{I}_{c}+b{I}_{s}=1;\text{}-a{I}_{s}+b{I}_{c}=0$

$I}_{c}=\frac{a}{{a}^{2}+{b}^{2}};\text{}{I}_{s}=\frac{b}{{a}^{2}+{b}^{2}$

This method relies only on the resource of contour integration as you asked!

Since our integrand is obviously holomorphic on the whole plane we get:

Breaking it into its three pieces we obtain:

Now let's call

and by solving:

This method relies only on the resource of contour integration as you asked!

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

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?