Complex differentiation

IsThisIt???

Studying for a Calculus repeat so will probably have loads of problems coming up but I'll start with this one.

The question is: at which points if any, is the function f(x+iy)=x^3 +iy^3 differentiable? And then determine the expression of f' at such points

What I have done so far is used the Cauchy-Riemann equations and found x^2=y^2 so this gave me x=y and x=-y.

However I am unsure of what this means. Are these the points at which the function is differentiable and are there more? Also how do I determine the expression of f' at such points?

Thanks very much for any help

Michael Collins

It means nothing!

[latex] \displaystyle x = \pm y. [/latex]

These form an 'x' like pattern in the complex plane.

To get the derivative you can use either of these, which hopefully you have seen before

[latex] \displaystyle \frac{\hbox{d}f}{\hbox{d}z} = \frac{\partial u}{\partial x} + \hbox{i} \frac{\partial v}{\partial x}[/latex]

[latex] \displaystyle \frac{\hbox{d}f}{\hbox{d}z} = \frac{\partial v}{\partial y} + \hbox{i} \frac{\partial u}{\partial y}[/latex]

where u is the real part of f, and y is the imaginary part of f.

EDIT:

But they do not satisfy the CR-equations on a domain (an open connected set). So the function is not holomorphic - see below.

IsThisIt???

So in calculating the derivative it doesn't matter if I give the answer as 3x^2 or 3y^2 since x=+/-y

Michael Collins wrote: »

It means the function f(z) is only complex differentiable on the lines

[latex] \displaystyle x = \pm y. [/latex]

.

Does this also mean the function is analytic on these lines?

Michael Collins

IsThisIt??? wrote: »

So in calculating the derivative it doesn't matter if I give the answer as 3x^2 or 3y^2 since x=+/-y

Actually a complex derivative doesn't exist for this function in an open set - see post below.

Does this also mean the function is analytic on these lines?

Nope, the function must satisfy the CR equations on an open connected set - any line in the complex plane is not an open set.

IsThisIt???

Once again, many thanks for your help.
Might pass this exam yet

Michael Collins

Actually, on further thought the function is not analytic! For this to be true it must satisfy the Cauchy-Riemann equations on an open set, which a line is not. So the function in question is not holomorphic (infintely differentiable on an open set) and hence not analytic. Sorry about the confusion.

Was this question on a past paper or something?

IsThisIt???

Yes it is from a past paper.

Does this mean the function is still differentiable on the lines x=+/- y but is not analytic since it is only differentiable on these lines and nowhere else.

For example a function such as f(x+iy)=3x + i3y, which would be differentiable everywhere, would be analytic??

TheBody

IsThisIt??? wrote: »

Yes it is from a past paper.

Does this mean the function is still differentiable on the lines x=+/- y but is not analytic since it is only differentiable on these lines and nowhere else.

For example a function such as f(x+iy)=3x + i3y, which would be differentiable everywhere, would be analytic??

I'm editing this to be a little more precise. The function IS differentiable along those lines but it is not analytic at any point because there is no neighbourhood about z in which the Cauchy-Riemann equations are satisfied.

TheBody

Can I ask what year in college you are in? I'm just curious what year they are teaching this stuff in.

IsThisIt???

I'm in 2nd year financial maths and economics in NUIG. These questions were taken from our exam this year

TheBody

You should see if your library has a book called "A first course in complex analysis with applications" by Dennis Zill and Patrick Shanahan. It's an excellent book to learn from.

IsThisIt???

I'll be sure to check it out next time i'm in galway. Thanks

Michael Collins

IsThisIt??? wrote: »

...For example a function such as f(x+iy)=3x + i3y, which would be differentiable everywhere, would be analytic??

Actually it's slighty more complicated than that. If f(z)

1) is continuous on the open set D, and
2) has partial derivatives of f(x+iy) with respect to x and y in D, and
3) satisfies the Cauchy-Riemann equations on an open set D,

then it is analytic.

In geneal the CR equations are necessary for f(z) to be analytic but not sufficient on their own.

But the example function you've given above does satisfy these three conditions, so yes, it is analytic.

TheBody wrote: »

Can I ask what year in college you are in? I'm just curious what year they are teaching this stuff in.

I would be curious to know this too, if the question asked really intends you to consider all the points above then it would seem to be quite a thorough course - not that the concepts are that hard, but a bit subtle perhaps.

Actually often these points are overlooked in simple text books and people often assume the CR equations are necessary and sufficient, which they are not.

TheBody

It's pretty involved stuff for a second year course. As said above by Michael, there is a lot going on here.

(P.S. I love complex analysis )