Complex Derivatives from frist principles

[NJunJie]

Homework Statement

How to prove there's a derivative from first principles?

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

z = complex = x + jy

It gets very completed.

Homework Equations

The Attempt at a Solution

 

[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??

 

[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?

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

 

[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.