# Thread: Which is Analytic

1. ## Which is Analytic

I have tried the 3 main options in pairs but didnt get it right.

2. ## Re: Which is Analytic

What do you mean by "three main options in pairs"? I interpret this as asking which of these would be possible "u" parts of a function for which some "v" will give an analytic function.

I presume you know that if u(x,y)+ iv(x,y) is analytic if and only if u and v satisfy the "Cauchy-Riemann" equations:
$\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$
$\displaystyle \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$

Now, notice what happens if we differentiate both sides of the first equation with respect to x and the second equation with respect to y:
$\displaystyle \frac{\partial^2 u}{\partial x^2}= \frac{\partial^2v}{\partial x\partial y}$
and
$\displaystyle \frac{\partial^2 u}{\partial y^2}= -\frac{\partial^2 v}{\partial x\partial y}$

That, is, u must satisfy
$\displaystyle \frac{\partial^2 u}{\partial x^2}= -\frac{\partial^2 u}{\partial y^2}$
or
$\displaystyle \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$
That is, a function, u(x,y) can be part of an analytic if and only if it satisfies that equation.

(Surely whoever gave you this problem expects you to know that.)

3. ## Re: Which is Analytic

Originally Posted by HallsofIvy
What do you mean by "three main options in pairs"? I interpret this as asking which of these would be possible "u" parts of a function for which some "v" will give an analytic function.

I presume you know that if u(x,y)+ iv(x,y) is analytic if and only if u and v satisfy the "Cauchy-Riemann" equations:
$\displaystyle \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$
$\displaystyle \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$

Now, notice what happens if we differentiate both sides of the first equation with respect to x and the second equation with respect to y:
$\displaystyle \frac{\partial^2 u}{\partial x^2}= \frac{\partial^2v}{\partial x\partial y}$
and
$\displaystyle \frac{\partial^2 u}{\partial y^2}= -\frac{\partial^2 v}{\partial x\partial y}$

That, is, u must satisfy
$\displaystyle \frac{\partial^2 u}{\partial x^2}= -\frac{\partial^2 u}{\partial y^2}$
or
$\displaystyle \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$
That is, a function, u(x,y) can be part of an analytic if and only if it satisfies that equation.

(Surely whoever gave you this problem expects you to know that.)
Yep just got that Q, can you help me with this?

http://oi41.tinypic.com/51r4u0.jpg

is the answer b for part 1?