Thread: How to use Cauchy Riemann

I have reached a step in the question where I have to apply the Cauchy Riemann formulas to:



How would I apply the Cauchy Riemann formula to answer the rest of the question?

And how would I know if it is analytic?

The cauchy-riemann formula:

The function f(z)= g(x,y)+ h(x,y)i is differentiable with respect to z= x+ iy only if it satisfies the "Cauchy-Riemann equations":



and

Originally Posted by zizou1089

I have reached a step in the question where I have to apply the Cauchy Riemann formulas to:



How would I apply the Cauchy Riemann formula to answer the rest of the question?

And how would I know if it is analytic?

The cauchy-riemann formula:

The function f(z)= g(x,y)+ h(x,y)i is differentiable with respect to z= x+ iy only if it satisfies the "Cauchy-Riemann equations":



and

You do precisely what you've wrote down. let = real part of your function i.e. and let = imaginary part of your function i.e.

Now it's just straight-forward partial differentiation

Find . This means you need to differenciate g w.r.t x.

so =

and as is the partial derivative of h w.r.t y

we get

Clearly

A function is analytic in the complex plane if the BOTH caucy riemann equations are satisfied. So since the first C-R equation i checked is not satisfied you can say that the function is not analytic