Using Cauchy's Tip for Changing Integral Variable Inconsistencies

[baby_1]

Homework Statement

I want to change integral variable to jx it means (w=jx)

Homework Equations

The Attempt at a Solution

but as you see the bounds of integral are different from the book text , what is my mistake?

 

[Simon Bridge]

hat about the part that says: "by using the Cauchy theorem and showing that..."?

 

[baby_1]

dear Simon
I didn't understand how can I use the Cauchy tip to solve this inconsistency?

 

[Simon Bridge]

It's a step you seemto have left out though.
I'll agree it looks like there's a minus sign missing. Maybe there's a typo? I can't actually read the image properly.

 

[pasmith]

dear Simon
I didn't understand how can I use the Cauchy tip to solve this inconsistency?

It allows you to perform the integral from $\omega = -i\infty$ to $\omega = +i\infty$, which on setting $\omega = ix$ becomes $x = -\infty$ to $x = \infty$.