Complex Derivatives from frist principles

1. Sep 2, 2009

NJunJie

1. The problem statement, all variables and given/known data

How to prove theres a derivative from first principles?

1/ [ z*sin(z)*cos(z) ]

z = complex = x + jy

It gets very completed.

2. Relevant equations

3. The attempt at a solution

2. Sep 2, 2009

Dick

Indeed, it gets very complicated. Sure you CAN prove it. Work out u(x,y) and v(x,y) and show they satisfy Cauchy-Riemann. That's equivalent to complex differentiability. But that's just silly. It's a colossal amount of work. z, cos(z) and sin(z) are differentiable. Show they are differentiable from first principles. Then just use the quotient rule and the product rule. Haven't we talked about this in another thread??

3. Sep 2, 2009

NJunJie

hi ya, thanks.
I've been trying and it just gets too complicated... :P
I've already proofed Cauchy Riemann for the denominator successfully - 2 pages long though.

Now i take the term alone and apply first prinicpes - just too COMPLEX. (well, i'll just state theorem as you mentioned - 'give up' heex).

Anyway, you've been a great help. :)

one more:-
sin (2z) / (z^15)

what is the location and nature of singularities here?

Answer: z=0 of order 15? (means 1 pole right? we always look at denominator alone?)

4. Sep 2, 2009

Dick

No, you don't just look at the denominator alone. E.g. z/z doesn't have a pole singularity. Expand the numerator in powers of z.