Determining Analyticity with Cauchy-Riemann Equations

[gtfitzpatrick]

Homework Statement

write down the C-R equations and use them to determine those points at which the following functions are analytic
(i)h(z)=$x^2-y^2-x + i(2xy+y)$
(ii)h(z)=cos2xcosh2y - isin2xsinh2y

Homework Equations

The Attempt at a Solution

ok so C-R equations are for z=u+iv eq1 = $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and eq2= $\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$

for for (i) i get $\frac{\partial u}{\partial x} = 2x-1 and \frac{\partial v}{\partial y} = 2x+1$
and $\frac{\partial v}{\partial x} = 2y and \frac{\partial u}{\partial y} = -2y$ equation 2 holds but equation 1 does not hold. So am i right in thinking the function is not analytic? both equations have to be satisfied right? its the way the question is worded is throwing me "those points" ...

 

[HallsofIvy]

A function is "analytic" at a given point if and only if it satisfies the C-R equation in some neighborhood of that point. Here, you are correct that the first function is not analytic for any points and the second function is analytic for all points.

 

[gtfitzpatrick]

and for part (ii) i got the reverse
as in $\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$
$\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$

which leads me to believe this function isn't analytic either...

 

[gtfitzpatrick]

Hi Hallsofivy,
Thanks a million for that. I didnt realize youd got in there between my question and me reply!
So I guess what your saying is it doesn't matter which equation has the negative?

 

[clamtrox]

So I guess what your saying is it doesn't matter which equation has the negative?

A function is analytic if (slightly hand-wavingly) it only depends on z, and not it's complex conjugate. You can see that in i), your function can be written as $f(z) = z^2 - \bar{z}$, meaning it's not analytic for sure.

If you have the C-R-equations holding with an additional minus sign, then the function only depends on complex conjugate of z (and not z itself). I think people call this kind of functions antianalytic.