Question

How to show a complex variable function is Analytic?
The question given is f (z) = e^(x^2−y^2)[cos(2 xy) + i sin( 2 xy)] is analytic, and find its derivative

Answer 1

Check it by Cauchy-Riemann condition :
\[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\]
and
\[\frac{\partial u}{\partial y}=\frac{-\partial v}{\partial x}\]

Answer 2

Isn't that the way to show that it is differentiable too? I'm a bit confused about it

Answer 3

how is "it is differentiable too"?

Answer 4

To show that a complex variable function has a derivative at a point we have to show that its real and imaginary parts are
continuously differentiable and satisfy the Cauchy Riemann equations
Is this correct?
If so how come it is the same method to show it is analytic too? Please explain I'm new to complex analysis.

Answer 5

yup, but Cauchy-Riemann is important, if you want to prove rigorous, you should use 2 conditions above. :)

Answer 6

So i've got to show that this function satisfies the Cauchy-Riemann equations. Thanks. :-D