Cauchy's integral formula

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Homework Statement



This problem is from Mary L. Boas - Mathematical Methods in the Physical Sciences, Chapter 14, section 3, problem 23.

[itex]\oint_C \frac{e^{3z}dz}{(z - ln2)^{4}}[/itex]

2. Homework Equations


Cauchy's integral formula

The Attempt at a Solution



First isolate the singularity:

[itex]\oint_C \frac{e^{3z}dz}{(z - ln2)^{4}}[/itex] (This should read something like (e^3z/1^4)/(z-ln2)^4) but I can't seem to do it in latex...)

Then let g(z) be:

[itex]g(z) = e^{3z}[/itex]

Since there's a fourth power in the singularity, g(z) must be derived three times:

[itex]g'(z) = 3e^{3z}[/itex]
[itex]g''(z) = 9e^{3z}[/itex]
[itex]g'''(z) = 27e^{3z}[/itex]


To solve the integral, we multiply g'''(z) by:

[itex] (2 \pi i)(27e^{(3)(ln2)}) = 432 \pi i[/itex]

But the book states that the answer is [itex]72 \pi i[/itex], which is exactly 6 times less than my answer. Where did I go wrong?
 
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vela
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Homework Statement



This problem is from Mary L. Boas - Mathematical Methods in the Physical Sciences, Chapter 14, section 3, problem 23.

[itex]\oint_C \frac{e^{3z}dz}{(z - ln2)^{4}}[/itex]

Homework Equations



Cauchy's integral formula

The Attempt at a Solution



First isolate the singularity:

[itex]\oint_C \frac{e^{3z}dz}{(z - ln2)^{4}}[/itex] (This should read something like (e^3z/1^4)/(z-ln2)^4) but I can't seem to do it in latex...)

Then let g(z) be:

[itex]g(z) = e^{3z}[/itex]

Since there's a fourth power in the singularity, g(z) must be derived three times:
You mean differentiated?

[itex]g'(z) = 3e^{3z}[/itex]
[itex]g''(z) = 9e^{3z}[/itex]
[itex]g'''(z) = 27e^{3z}[/itex]

To solve the integral, we multiply g'''(z) by:

[itex] (2 \pi i)(27e^{(3)(ln2)}) = 432 \pi i[/itex]

But the book states that the answer is [itex]72 \pi i[/itex], which is exactly 6 times less than my answer. Where did I go wrong?
To calculate the residue, you have to divide by 1/(n-1)! where n is the order of the pole. It's part of the limit formula for calculating the residue.
 
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Thank you very much for your answer.
 

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