Finding Local Max/Min of Complex Function

[sara_87]

Homework Statement

How can i determine whether a complex function has any local maximum or minimum?

Homework Equations

let's consider the case f(z)=z* (conjugate of z)
z=x+iy

The Attempt at a Solution

f(z)=z*=x-iy

how do i see if it has local max or min?

Thank you

 

[Mark44]

$$f(z)~=~z\bar{z}~=~(x + iy)(x - iy)~=~x^2 + y^2$$
So f(z) is purely real, for any complex number z. It looks to me like it has an absolute minimum but no maximum.

 

[sara_87]

how do you know it has an absolute minimum?

 

[Mark44]

x and y are real numbers. Both x² and y² are always nonnegative, so their sum will also always be nonnegative.

 

[sara_87]

i agree, but why does this have anything to do with max/min?

 

[Mark44]

The minimum value of your f(z) is 0 and there is no maximum value. Are you asking about complex functions in general, or about this one?

 

[sara_87]

im asking in general.
if we look at your example, f(z)=zz*
then why is there a minimum?

 

[Mark44]

If you're asking in general, then you would be looking at the derivative and seeing where it's zero, and testing the critical points.

You're overthinking your example (it's not mine). You have f(z) = zz* = x² + y², where z = a + bi. For this particular function, it's very easy - almost trivial - to discover that the minimum value is 0 (for z = 0 + 0i), and that the function is unbounded. For any real numbers x and y, x² $\geq$ 0 and y² $\geq$ 0, which means that x² + y² $\geq$ 0.

If you think about it graphically, the complex plane is the domain and the image of the function is a paraboloid that opens upward and whose vertex is at (0, 0).

 

[sara_87]

oh, so the graph would be like y=x^2 in real coordinate system
?

how did you know that f=x^2+y^2 looks like that in the complex plane?

 

[Mark44]

oh, so the graph would be like y=x^2 in real coordinate system
?

how did you know that f=x^2+y^2 looks like that in the complex plane?

Not in the complex plane. The complex plane is the domain.
I know what it looks like because I know what the graph of z = x² + y² looks like in R³. The only thing different is that the domain in one of these is the complex plane and in the other it's the real x-y plane.

 

[sara_87]

ok, so it's in 3d. I get it now, so we have: z=x^2+y^2 not f(z)=x^2+y^2
right?

 

[LCKurtz]

In the original post, you asked "How can i determine whether a complex function has any local maximum or minimum?".

In general, the question doesn't make any sense because the complex numbers aren't ordered. Which is larger, 3 + 4i or 4 + 3i?

You have given a particular function that happens to have real values, which is why your particular problem makes sense.