How do you show that a complex function is analytical?

[twotaileddemon]

Like if I wanted to show how sin z, cos z, or e^z are analytical, what is the general process I have to do? Can I use the cauchy - riemann relations somehow?

(where z = x + iy is complex)

 

[LCKurtz]

Yes. For example, e^{z =} e^x+iy = e^xe^iy =e^x(cos(y) + i sin(y)) so

u = e^xcos(y) and v = e^xsin(y)

and you can Cauchy-Riemann away.

 

[twotaileddemon]

Thanks! :)