Sinh series manipulation

  • Thread starter saybrook1
  • Start date
  • #1
101
4

Homework Statement


Hello, I'm not trying to solve this exact problem although mine is similar and I am confused on how they were able to get a -1 in the exponent from one step to another.

Homework Equations


I have attached a picture indicating the step that I am confused about. How are they able to manipulate the series and pull out that -1 into the exponent thereby finding the residue?

The Attempt at a Solution


Some sort of series manipulation that I can't figure out; any help is greatly appreciated, thank you guys!
 

Attachments

  • example1.png
    example1.png
    28.4 KB · Views: 324

Answers and Replies

  • #2
277
97
It comes from the reciprocal rule for exponents.

In general, ## \frac{1}{x^n} = x^{-n} ##. So as an example:

## \frac{1}{x^4-x^6} = \frac{1}{x^4(1-x^2)} = x^{-4}(1-x^2)^{-1}##
 
  • #3
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,796
1,669

Homework Statement


Hello, I'm not trying to solve this exact problem although mine is similar and I am confused on how they were able to get a -1 in the exponent from one step to another.

Homework Equations


I have attached a picture indicating the step that I am confused about. How are they able to manipulate the series and pull out that -1 into the exponent thereby finding the residue?

The Attempt at a Solution


Some sort of series manipulation that I can't figure out; any help is greatly appreciated, thank you guys!
The common factor -(z - πi)3 can be factored out of the series expression of sinh3 z before it is inverted. The series involves only alternating odd powers of the common factor. After that, for the inversion, one can apply mfiq's hint about using the law of exponents.

There's no magic here - just straightforward algebra.
 
  • #4
101
4
Awesome, makes perfect sense. Thank you both.
 
  • #5
101
4
If either of you can still see this, would you possibly be able to tell me how they then lose that -1 power on the next line down allowing them to find the residue? Thanks again.
 
  • #6
101
4
If either of you can still see this, would you possibly be able to tell me how they then lose that -1 power on the next line down allowing them to find the residue? Thanks again.
Nevermind, think I've got it now.
 

Related Threads on Sinh series manipulation

  • Last Post
Replies
2
Views
6K
  • Last Post
Replies
5
Views
777
Replies
7
Views
5K
  • Last Post
Replies
4
Views
912
Replies
12
Views
11K
Replies
0
Views
892
  • Last Post
Replies
3
Views
700
  • Last Post
Replies
4
Views
911
  • Last Post
Replies
2
Views
3K
Top