Residue of Function f(z)=e1/z/(1-z): Guide and Explanation

[aiisshsaak]

Hello guys,
I just want to confirm about this problem ..Find the residue of this function: f(z)=e^1/z/(1-z)

Thx in advance.

 

[HallsofIvy]

"Confirm"? You mean you have already found an answer? Great! Tell us what you got and we will try to confirm it for you.

 

[aiisshsaak]

Right :p
the answer should be "exp"?

 

[Office_Shredder]

The answer should be a number, not a function.

Also, there are two different points where you might be interested in the residue - at z=0 and z=1. Which one are you supposed to find?

 

[aiisshsaak]

The answer should be a number, not a function.

Also, there are two different points where you might be interested in the residue - at z=0 and z=1. Which one are you supposed to find?

at z_°=1

here is how I've done it ..

Res=limit [f(z)*(z_°-z)] as z goes to z_°
=limit [e^1/z/[STRIKE](1-z)[/STRIKE]*[STRIKE](z_°-z)[/STRIKE]] as z goes to z_°
=limit [e^1/z] as z goes to z_° ''which equals to 1''
and finally I plugged in z=z_°=1 and I got: Res=e^1/1=e≈2.72

am I wrong somewhere?

 

[Office_Shredder]

That looks correct except that it should be f(z)*(z-z₀) which means you're off by a minus sign.

Usually "exp" refers to the function e^x, not the number e itself, which is why I wrote the first part of my previous post.

 

[aiisshsaak]

Thx guys for these replies..I just want to know one last thing. using the residue theorem, what is the integral of f(z) ...at |z|=1/2

Thx again :)

 

[Office_Shredder]

Your question isn't very well posed... Are you integrating around the circle defined by |z|=1/2?

 

[aiisshsaak]

Your question isn't very well posed... Are you integrating around the circle defined by |z|=1/2?

Im sry about that, yes integration around the circle .

 

[aiisshsaak]

anyone can help me with the integration please? f(z)=e^1/z/(1-z)
∫f(z)dz around the circle |z|=1/2 ?

 

[daveyrocket]

You need to identify the poles within the circle |z| = 1/2. You might consider expanding the exponential in a series.

 

[aiisshsaak]

You need to identify the poles within the circle |z| = 1/2. You might consider expanding the exponential in a series.

All I need is the result, so can u provide me with that? thx

 

[Office_Shredder]

All I need is the result, so can u provide me with that? thx

No, we can't. We can help guide your work but we will not tell you what the answer is.

There is a single pole at z=0 inside of the circle (it should be fairly obvious from looking at the function). Can you find the residue at that pole?

 

[aiisshsaak]

No, we can't. We can help guide your work but we will not tell you what the answer is.

There is a single pole at z=0 inside of the circle (it should be fairly obvious from looking at the function). Can you find the residue at that pole?

But I thought the pole is z=1, no?

 

[daveyrocket]

This function has more than one pole. Look at the 1/z in the exponential.

 

[aiisshsaak]

This function has more than one pole. Look at the 1/z in the exponential.

Right, ty person ;)