How can I calculate the residue at 0 of the next function:
??
Hope you'll be able to help me
Thanks a lot!
By calculating the residue "at infinity" of .
That is, write sin(z) as a Taylor's series, then multiply by . That, of course, will be a power series with only positive powers. Now invert the powers getting a Laurent series with only negative powers. The residue is the coefficient of the term.
Let's start with the series expansion of the function...
(1)
... from the well know result...
(2)
The most 'spontaneous' approach is to impose...
(3)
... and that permits us to compute the in recursive way...
... and so one...
Now we can write...
(4)
... from which it follows...
(5)
... so that the residue of in is ...
Kind regards