Find the maximum value of this complex function

[supermiedos]

Homework Statement

Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1

Homework Equations

|f(z)| yields the maximum value

The Attempt at a Solution

f(z) = exp(x) ( cosy + i siny)

Unfortunately that's all I've got. I've seen examples with polynomials, but not with trigonometric functions. Please, any help?

 

[RUber]

What is the maximum value of just the real part if you have |real(z) - 1 | < 1?
Does the imaginary part of the exponential function impact its modulus? i.e. what is | e^ix | ?
$e^{a+ib} = e^a e^{ib}$
Start with those...you should get a sense of what needs to happen.

 

[exclamationmarkX10]

The maximum value of f(z) doesn't make sense since f(z) is a complex number. In general, can one complex number be considered less than another complex number? e.g. is 3i "less than" 4i? These numbers cannot be compared this way however real numbers can.

Have you learned the maximum modulus principle? If yes, you can parametrize the circle, plug that into the function and then evaluate the modulus. The maximum value will then become clear.

 

[Ray Vickson]

Homework Statement

Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1

Homework Equations

|f(z)| yields the maximum value

The Attempt at a Solution

f(z) = exp(x) ( cosy + i siny)

Unfortunately that's all I've got. I've seen examples with polynomials, but not with trigonometric functions. Please, any help?

Try writing $z = 1+i+w$, with $w = x + i y$ and where $x,y$ are real. What do your restrictions on $z$ become in the new variables $w$ or $x,y$? What does $|f(z)|$ look like in the new variables?

 

[Rellek]

Hi,

I think the most convenient way to start is to imagine the triangle inequality for your boundary. As someone has pointed out already, what would the modulus for exp(ix) be? This will help because the largest modulus of z that you can find will also happen to be the largest modulus for exp(z).

I think a good place to start is to somehow use the triangle inequality on this rather simple identity:

|z| = |z - (1+i) + (1+i)|, and see if you can make use of the transitive property for inequalities.

 

[HallsofIvy]

Do you not know that "if a complex valued function is analytic on a set, then it takes on maximum and minimum values only on the boundary of the set.''

 

[Ray Vickson]

Hi,

I think the most convenient way to start is to imagine the triangle inequality for your boundary. As someone has pointed out already, what would the modulus for exp(ix) be? This will help because the largest modulus of z that you can find will also happen to be the largest modulus for exp(z).

I think a good place to start is to somehow use the triangle inequality on this rather simple identity:

|z| = |z - (1+i) + (1+i)|, and see if you can make use of the transitive property for inequalities.

If the OP pays attention to post #4, the problem becomes elementary and needs little, if any, deep properties.