Jordan's Lemma technicalities - where does sin(mz) go

[thomas49th]

Homework Statement

I don't understand why it's so great m>0 here

http://gyazo.com/cfa21b39653275a38ec80722b7faa58b
From these lectures notes (I think they are generally pretty good)

Where has the sin(mz) gone?

When going from 4.67 to 4.68 has

e^imz = cos(mz) + isin(mz) been used? I'm guessing that was the trick, but I don't see where sin(mz) has pootled off to?

 

[Mark44]

Homework Statement

I don't understand why it's so great m>0 here

http://gyazo.com/cfa21b39653275a38ec80722b7faa58b
From these lectures notes (I think they are generally pretty good)

Where has the sin(mz) gone?

When going from 4.67 to 4.68 has

e^imz = cos(mz) + isin(mz) been used?

Yes.

I'm guessing that was the trick, but I don't see where sin(mz) has pootled off to?

Instead of looking at $\int \frac{cos(mx)dx}{a^2 + x^2}$, the writer of the notes is looking at $\int \frac{e^{mz}dz}{a^2 + z^2}$ and getting a value. He also notes that the complex part, which contains the sin(mx) numerator, turns out to be zero, due to the fact that the integrand is an odd function. The integral of an odd function over (-∞, ∞) is zero.

 

[Ray Vickson]

Homework Statement

I don't understand why it's so great m>0 here

http://gyazo.com/cfa21b39653275a38ec80722b7faa58b
From these lectures notes (I think they are generally pretty good)

Where has the sin(mz) gone?

When going from 4.67 to 4.68 has

e^imz = cos(mz) + isin(mz) been used? I'm guessing that was the trick, but I don't see where sin(mz) has pootled off to?

At the bottom he says that the sin(mx) integral vanishes, leaving only the cos(mx) part. It says that right in the document!

RGV

 

[thomas49th]

I think I'm happy with the sin(mx) disappearing. The m variable is causing me grief now:

So for the variable m, what decides it. If I'm given a functions like

1/(x+1)^4

is the variable m in the e^imz equal to the overall degree of the function

ie would m=2 if

x^2/(x+1)^4 was my function?

Thanks
Thomas

 

[Mark44]

Neither m nor a is a variable - both are constants as far as the integration is concerned. The only variables are x and z.