- #1

xspook

- 19

- 0

## Homework Statement

What is the integral from negative infinity to positive infinity of the following functions?

a) f(z) = [itex]\frac{e^{-i5z}}{z^{2}+1}[/itex]

b) f(z) = [itex]\frac{e^{-i5z}}{z^{2}-1}[/itex]

c) f(z) = [itex]\frac{1}{π}[/itex][itex]\frac{a}{z^{2}+a^{2}}[/itex]

d) f(z) = [itex]e^{\frac{-(z-ia)^{2}}{2}}[/itex]

e) f(z) = [itex]\frac{sinz}{z}[/itex]

## Homework Equations

All my professor showed was an example, which would be (c) above

[itex]\frac{1}{π}[/itex][itex]\int\frac{a}{z^{2}+a^{2}}[/itex]dz

[itex]\frac{1}{π}[/itex][itex]\int\frac{1}{2ai}[/itex]([itex]\frac{1}{z-ia}[/itex]-[itex]\frac{1}{z+ia}[/itex])dz

[itex]\frac{1}{2iπ}[/itex][itex]\int\frac{1}{z-ia}[/itex]-[itex]\frac{1}{z+ia}[/itex] dz=1

dz[itex]\Rightarrow[/itex][itex]Re^{iθ}idθ[/itex]

limits of integration are now from 0 to π

[itex]\int\frac{1}{R^{2}e^{2iθ}+a^{2}}[/itex][itex]Re^{iθ}iθdθ[/itex]

[itex]R^{2}[/itex][itex]\rightarrow[/itex]∞

[itex]\oint[/itex]f(z)dz=0

f(z) is analytic

f(z) = [itex]\frac{1}{2πi}[/itex]

and this is where the class time expired and he said "you can figure out the rest"

## The Attempt at a Solution

I have no clue where to start with these. The class is Mathematical Methods of Physics and this year there weren't enough students in the class so they offered it as an independent study. The professor that "observes" our work didn't tell us how to do these or where to start. I'm just looking for guidance on what to do, that an idiot can understand. Often I read threads on here and the way some of you convey your knowledge is out of my league. I would appreciate any help! Thanks