Complex conjugate

[Mondo]

Hello,

I have a problem understanding an example of an integral of complex conjugate from the book Visual Complex analysis. Here is the fragment that confuses me:

I understand the equation marked by me by red 1. However I don't get how they transformed the Im on left hand side to i on THS in (8)? Also I have stressed by a red line the fragment that says "we would multiply by r" where is it in 8? Looks like no multiplication has happened. Finally I don't see how they prove the above integral is purely imaginary - what happened to the real part?

Thanks.

 

[Dr.Peterson]

Hello,

I have a problem understanding an example of an integral of complex conjugate from the book Visual Complex analysis. Here is the fragment that confuses me:
View attachment 30959
I understand the equation marked by me by red 1. However I don't get how they transformed the Im on left hand side to i on THS in (8)? Also I have stressed by a red line the fragment that says "we would multiply by r" where is it in 8? Looks like no multiplication has happened. Finally I don't see how they prove the above integral is purely imaginary - what happened to the real part?

Thanks.

It appears that what is said about multiplying by r is about comparing this problem to [12], which you haven't shown us, so it's hard to comment:

​

Some reasoning there appears to be the basis for the conclusion that the integral is purely imaginary. Can you show it to us (both the picture and the explanation)?

 

[blamocur]

It appears that what is said about multiplying by r is about comparing this problem to [12], which you haven't shown us, so it's hard to comment:

View attachment 30961​

Some reasoning there appears to be the basis for the conclusion that the integral is purely imaginary. Can you show it to us (both the picture and the explanation)?

Just a guess: if [12] showed that [imath]\oint \frac{dz}{z}[/imath] is purely imaginary, then the same is true of [imath]\oint \bar z dz[/imath] because the ratio of [imath]\bar z[/imath] and [imath]\frac{1}{z}[/imath] is purely real, i.e. [imath]|z|^2[/imath]. I am also guessing that the integration is over a circle where [imath]|z|[/imath] is constant.

 

[Mondo]

Below are two pages describing the problem [12]



Do they claim that the result is purely imaginary because the horizontal component vanished? This can be read from the description below figure [13].

Just a guess: if [12] showed that [imath]\oint \frac{dz}{z}[/imath] is purely imaginary, then the same is true of [imath]\oint \bar z dz[/imath] because the ratio of [imath]\bar z[/imath] and [imath]\frac{1}{z}[/imath] is purely real, i.e. [imath]|z|^2[/imath]. I am also guessing that the integration is over a circle where [imath]|z|[/imath] is constant.

Can you elaborate why do you think the ratio between [imath]\bar z[/imath] and [imath]\frac{1}{z}[/imath] is purely real?

Thanks