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Complex numbers
Complex numbers  
QUESTION: Dear Dr. Patton, I'm a bit stuck with complex numbers. In this exercise (see attachment) I am asked to simplify each of the following expressions to either the standard or exponential form.
I'm a bit confused and I don't know where I should start from.

I look forward to hearing from you.

Thanks in advance

Best regards

complex algebra
complex algebra  
ANSWER: Expressions with complex numbers look, well, complicated, but they are easy to manipulate. For your problem you need to use just a few straightforward definitions, including:

Combining exponents:

(x^n)(x^m) = x^(n+m)

(x^n)^m = x^(nm)

(x^n)/(x^m) = (x^n)(x^(-m)) = x^(n-m).

Complex number relations (where j = sqrt(-1)):

for z = x + iy = complex number we have Re(z) = x and Im(z) = y

e^(jx) = cosx + isinx

(e^jx)^n = e^(jxn) so that (cosx + isinx)^n = cos(xn) + isin(xn)    <--  this is a biggy (de Moivre's thm).

Using these relations, you can simplify the expressions as in the attached image.

BTW, here are 2 other relations you should keep in your quiver:

cosx = (e^jx + e^-jx)/2  &  sinx = (e^jx - e^-jx)/2i.

Randy

---------- FOLLOW-UP ----------

QUESTION: Dear Dr. Patton, first of all thanks for your extremely quick reply. Just another question, when you say "you can simplify the expressions as in the attached image", is there supposed to be any image attachment? I couldn't find nothing.
Sorry for this silly question.

Best regards

Stefano

Answer
complex algebra
complex algebra  
Not a silly question. I did attach an image with the "solutions" in my previous answer so I am wondering what happened to it. I checked the "Answered Questions" selection and it was there, along with your original image.

Whatever, I'll attach the image again with the website SW. Let me know if you don't receive it.

Randy

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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