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## Main Question or Discussion Point

This is related to my thread on 'integrals over an infinite range (using residue theorem)'. I am trying to understand how certain integrals over semi circles, or the widths of rectangles go to zero under certain functions.

I want to evaluate the integral [tex]\int\frac{exp(ax)}{coshx}dx[/tex] from -infinity to plus infinity.

So consider [tex]\oint\frac{exp(az)}{coshz}dz[/tex] around a rectangle. This is equal to

[tex]\int\frac{exp(ax)}{coshx}dx[/tex] from R to -R

PLUS

[tex]\int\frac{exp(a(i*pi+x))}{coshx}dx[/tex] from R to -R (contains one singularity)

PLUS two integrals corresponding to the sides. They look like this:

I=[tex]\int\frac{exp(aR+ait)}{cosh(R+it)}idt[/tex] from 0 to pi. Now these integrals apparently EQUAL ZERO when R tends to infinity. Can anyone help me see why this is?

Perhaps I could use ML inequality but how?

I want to evaluate the integral [tex]\int\frac{exp(ax)}{coshx}dx[/tex] from -infinity to plus infinity.

So consider [tex]\oint\frac{exp(az)}{coshz}dz[/tex] around a rectangle. This is equal to

[tex]\int\frac{exp(ax)}{coshx}dx[/tex] from R to -R

PLUS

[tex]\int\frac{exp(a(i*pi+x))}{coshx}dx[/tex] from R to -R (contains one singularity)

PLUS two integrals corresponding to the sides. They look like this:

I=[tex]\int\frac{exp(aR+ait)}{cosh(R+it)}idt[/tex] from 0 to pi. Now these integrals apparently EQUAL ZERO when R tends to infinity. Can anyone help me see why this is?

Perhaps I could use ML inequality but how?