Expressing a complex function as polar coordinates

[PedroB]

Homework Statement

Consider the complex function f (z) = (1 + i)^z with z ε ℂ.

1. Express f in polar coordinates.

Homework Equations

The main derived equations are in the following section, there is no 'special rule' that I (to my knowledge) need to apply here.

The Attempt at a Solution

I calculated that the equation is equal to (√2)(e^x)(cos(y)+icos(y)+isin(y)-sin(y))
or, alternatively (e^((xln(√2)-y∏/4))(e^(i∏((x/4)+yln(√2))

(These expansions may not be correct though)

My main problem lies in the question itself. Even though I have done these expansions I have no idea whether or not they are relevant for the problem at hand (The fact that it is polar coordinates that the question is after makes me believe that the latter expansion is irrelevant), and if they are I am unsure as to how to represent them as 'polar coordinates'. Any help would be greatly appreciated, thank-you.

 

[tiny-tim]

Hi PedroB!

Hint: what is 1 + i in polar notation? ​

 

[PedroB]

If I express (1+i)^z by substituting the (1+i) equivalence in polar form I simply get (cos(z∏/4)+isin(z∏/4)). What I don't understand is if by expressing it in polar coordinates they mean for me to actually plot the graph on the complex plane. If this is the case I still do not see how I can get an answer (Thanks for the first tip though)

 

[haruspex]

If I express (1+i)^z by substituting the (1+i) equivalence in polar form I simply get (cos(z∏/4)+isin(z∏/4)).

That can't be right because 1+i has modulus > 1. Besides, that isn't polar form. If you get it into the complex form re^iθ then the polar form is (r,θ).

 

[PedroB]

My mistake, I apologize, the polar form will therefore be ((√2)^z,(z∏/4)). Is this it then? It seems too simple (from experience of the questions I've been given in the past), but if this is truly expressing the equation in polar coordinate form is there nothing more I can do to it? Would it be sufficient to leave z as it is?

 

[tiny-tim]

My mistake, I apologize, the polar form will therefore be ((√2)^z,(z∏/4)).

if z ε ℝ, yes

but the question says z ε ℂ ​

(and now I'm off to bed :zzz:)

 

[haruspex]

Sorry, I was careless in the wording of my previous post.
re^iθ is "polar form", but it's the polar form of a complex number. The question asks for polar co-ordinates, which is (r,θ) form, and does not involve complex numbers.
So here's what I should have written:

If you get it into the complex polar form, re^iθ, then the polar co-ordinate form is (r,θ).​

the polar form will therefore be ((√2)^z,(z∏/4)).

No, z is complex. You need the r and θ terms to be real.
Start with expressing z as x+iy and put 1+i in re^iθ form. Turn the resulting expression into re^iθ form.