Contour Integrals

May 2010
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0
I'm working on a problem that has contour integrals and checked the answers my professor provided with a solution sheet, however there's a part in the solution sheet I don't understand.

This is the question given:


This is the solution:




And this is the part I don't understand from this solution:



Thanks in advance for any help.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
I'm working on a problem that has contour integrals and checked the answers my professor provided with a solution sheet, however there's a part in the solution sheet I don't understand.

This is the question given:


This is the solution:




And this is the part I don't understand from this solution:



Thanks in advance for any help.
Do you think possibly it was the "i" that was already in the integral: \(\displaystyle (cos^2 \theta)(\underline{i} e^{i\theta})\)?
 
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May 2010
2
0
Do you think possibly it was the "i" that was already in the integral: \(\displaystyle (cos^2 \theta)(\underline{i} e^{i\theta})\)?
Yes, that's what I think it is, and if that's the case there shouldn't be i^2 later on etc... I'm assuming it's probably just a typo? I think it's best if I directly contact my professor about this.
 
Apr 2010
384
153
Canada
Do you think possibly it was the "i" that was already in the integral: \(\displaystyle (cos^2 \theta)(\underline{i} e^{i\theta})\)?
I don't think this is the case (though it appears likely),

\(\displaystyle \int ( cos^2 \theta ) ( i e^{ i \theta } ) d \theta \)

We need to get to the above from

\(\displaystyle i \int ( \frac{1}{2} [ e^{i \theta } + e^{-i \theta} ]^2 )( i e^{ i \theta} ) d\theta \)

Let's look at

\(\displaystyle \frac{1}{2} [ e^{i \theta } + e^{-i \theta} ]^2 \)

\(\displaystyle ( \frac{1}{2} [ e^{i \theta } + e^{-i \theta} ] )^2 = ( \frac{1}{2} [ (cos \theta + i sin \theta ) + (cos \theta - i sin \theta) ] )^2 \)

\(\displaystyle ( \frac{1}{2} ( 2 cos \theta ) )^2 = cos^2 \theta \)

Which transforms

\(\displaystyle i \int ( \frac{1}{2} [ e^{i \theta } + e^{-i \theta} ]^2 )( i e^{ i \theta} ) d\theta = i \int cos^2 \theta (i e^{i \theta} ) d \theta \)

We have an extra i here....I don't know where it came from but I'm relatively sure it didn't come from what was already in the integral.
 
Last edited:

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Yes, that's what I think it is, and if that's the case there shouldn't be i^2 later on etc... I'm assuming it's probably just a typo? I think it's best if I directly contact my professor about this.
Oh, I see your point- the i also remained inside the integral! Yes, it most likely a typo- they intended to take the i outside the integral but accidently also left it inside.