The Complex Exponential And Regions in the Complex Plane

Hello all, I have done a few questions that I need corrected.

1.) Sketch the region of the complex plane defined by Im (z^2) = 2 for pi/8 < arg (z) < 3pi/8

my solution: I let z = x + iy, then found z^2 and the imaginary parts of z^2 which gave me a graph shape of 1/x. the region defined by the angle is in the first quadrant and only a small bit of that curve is included.

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2.) z = 1/4 - [root(3) / 4]i

a) Write z in exponential polar form for angles between -pi and pi

my solution: 1/2 * e ^ i (pi/3)

b) Calculate (1/z) ^ 9

my solution: -2 * 2^8 + 0i

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3.) Use the complex exponential to express sin^3(theta) as a sum of sines or cosines of multiples of theta.

My solution: -1/i sin(3 theta) + 3/i cos (theta)

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Thank you in advance.

Re: The Complex Exponential And Regions in the Complex Plane

Quote:

Originally Posted by

**andrew2322** Hello all, I have done a few questions that I need corrected.

1.) Sketch the region of the complex plane defined by Im (z^2) for pi/8 < arg (z) < 3pi/8

my solution: I let z = x + iy, then found z^2 and the imaginary parts of z^2 which gave me a graph shape of 1/x. the region defined by the angle is in the first quadrant and only a small bit of that curve is included.

__________________________________________________ ______________________

2.) z = 1/4 - [root(3) / 4]i

a) Write z in exponential polar form for angles between -pi and pi

my solution: 1/2 * e ^ i (pi/3)

b) Calculate (1/z) ^ 9

my solution: -2 * 2^8 + 0i

__________________________________________________ ______________________

3.) Use the complex exponential to express sin^3(theta) as a sum of sines or cosines of multiples of theta.

My solution: -1/i sin(3 theta) + 3/i cos (theta)

__________________________________________________ ______________________

Thank you in advance.

1. Are you sure it didn't say $\displaystyle \displaystyle \frac{\pi}{8} < \arg{\left(z^2\right)} < \frac{3\pi}{8}$

2. a) It's clear that $\displaystyle \displaystyle z = \frac{1}{4} - \frac{\sqrt{3}}{4}i$ is in the fourth quadrant, so $\displaystyle \displaystyle \arg{(z)}$ can not possibly be $\displaystyle \displaystyle \frac{\pi}{3}$.

b) This is easiest to evaluate if you have evaluated part a) correctly...

Re: The Complex Exponential And Regions in the Complex Plane

Hello Prove it. I checked the question again and it says arg (z) not arg (z^2)

also, for part 2a) I realise my careless mistake, I was meant to type - pi/3.

due apologies.

Re: The Complex Exponential And Regions in the Complex Plane

Sorry Prove It and others. The Im (z^2) = 2, i forgot to include the = 2 part. very sorry.

I have edited the original post to include this correction.

Re: The Complex Exponential And Regions in the Complex Plane

Quote:

Originally Posted by

**andrew2322** Hello Prove it. I checked the question again and it says arg (z) not arg (z^2)

also, for part 2a) I realise my careless mistake, I was meant to type - pi/3.

due apologies.

Then $\displaystyle \displaystyle z = \frac{1}{2}e^{-\frac{\pi i}{3}}$ is correct.

Re: The Complex Exponential And Regions in the Complex Plane

Re: The Complex Exponential And Regions in the Complex Plane

Quote:

Originally Posted by

**andrew2322** is part B correct?

You tell me. $\displaystyle \displaystyle z = \frac{1}{2}e^{-\frac{\pi i}{3}}$, so what is $\displaystyle \displaystyle z^{-9}$?

Re: The Complex Exponential And Regions in the Complex Plane

Re: The Complex Exponential And Regions in the Complex Plane

which is equivalent to 2^9 e^pi which is 2^9 (cos(pi) +isin(pi)) which is -1 * 2^9?

Re: The Complex Exponential And Regions in the Complex Plane

Quote:

Originally Posted by

**andrew2322** 2^9 e ^3pi?

Correct, but don't forget the 'i':

$\displaystyle \left(\frac{1}{2}e^{-\frac{\pi i}{3}}\right)^{-9}=2^9e^{3\pi i}$

Re: The Complex Exponential And Regions in the Complex Plane

Cool, thanks for the heads up. I really have to stop doing that haha.

Would you happen to have any clue on how to do Q1)? I reposted it because this one got quite messy and still have yet to receive a response.