Power Series Expansion and Residue Calculation for log(1-z)

[squenshl]

Homework Statement

Find a power series expansion for log(1-z) about z = 0. Find the residue at 0 of 1/-log(1-z) by manipulation of series, residue theorem and l'hospital's rule.

Homework Equations

The Attempt at a Solution

Is this power series the same as the case for real numbers.

 

[squenshl]

I have the power series expansion about z = 0 for log(1-z).
-z - z²/2 - z³/3 - ...
But how do I find the residues with the methods mentioned

 

[squenshl]

When I manipulate do I use the power series for log(1-z)

 

[squenshl]

My series for -1/log(1-z) is:
1/z - 1/2 - z/12 - z²/24 - ...
So my residue is a_-1 = -1/2.
Is that right?

 

[squenshl]

How do I do it by the residue theorem and l'hospital's rule.

 

[Susanne217]

Make me think of the power series of log(1+z)

Recall Sir,

$$log(1+z) = \sum_{j=1}^\infty} \frac{(-1)^{j+1}}{j}z^{j} = z - \frac{z^2}{2} + \frac{z^3}{3}-\cdots$$

so by very very simply replacing z with -z

you get $$-z - \frac{(-z)^2}{2} + \frac{(-z)^3}{3}-\cdots$$

So the power series expansion of log(1-z)

Is $$P_{n} = -\sum_{j=1}^{n} \frac{z^n}{n}$$

 

[vela]

My series for -1/log(1-z) is:
1/z - 1/2 - z/12 - z²/24 - ...
So my residue is a_-1 = -1/2.
Is that right?

No. Where'd you get -1/2 from?

 

[squenshl]

The second term in the series.

 

[squenshl]

I think I got it now. It is 1 beacuse this is the constant for the z^-1 term (the term 1/z)

 

[squenshl]

Using the formula for the residue at a simple pole (Residue theorem) I also get 1 as my residue.
res₀ = 1

 

[vela]

Yup, that's right.

When you say "residue theorem," you're not referring to the Cauchy residue theorem, right? Because that's what most people mean by that term.

http://mathworld.wolfram.com/ResidueTheorem.html

 

[squenshl]

True.
Also got 1 using l'hospital's rule.
Didn't realize it was so easy.
Cheers.

 

[twoforone]

Hi Every body!

I wan to compute the power series expansion of dedekind eta function. Specifically, I want to know the power series expansion of η(τ)/η(3τ)? How could I expand this function? I would be happy if you could help me as I am stuck at this state when I am computing the modular polynomial of prime number 3.