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September 7th, 2010, 08:50 PM   #1
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Express in a+bi form

Define:


Evaluate:


My answer (is there a way to make these symbols larger?):


Where the ellipses (...) means "What comes next?" or "Is this right so far?"

Thanks.
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September 8th, 2010, 03:49 AM   #2
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Re: Express in a+bi form

Same exact post, but a little bit more readable.

Define:


Evaluate:


My answer (is there a way to make these symbols larger?):


Where the ellipses (...) means "What comes next?" or "Is this right so far?"

Thanks.
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September 10th, 2010, 06:49 PM   #3
 
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Re: Express in a+bi form

Hi,
I am just thinking about the word "evaluate"... does it mean simplify or something? In which sence? I found the original expression pretty simple...

Best regards,
Jens
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September 10th, 2010, 07:04 PM   #4
 
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Re: Express in a+bi form

Replying myself... just an idea. Why you don't use the normal decomposition

I mean, looks not that bad ...and again, me idiot: I should have read the header in this case. Damn...
Best regards,
Jens
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September 11th, 2010, 10:22 AM   #5
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Re: Express in a+bi form

Quote:
Originally Posted by MSUMathGuy
Same exact post, but a little bit more readable.

Define:


Evaluate:


My answer (is there a way to make these symbols larger?):


Where the ellipses (...) means "What comes next?" or "Is this right so far?"

Thanks.
The goal was to keep the answer as a complex number in the form of a+bi.

The trick was to use Euler's formula, which says:


The other trick was to work out this case of Euler's formula:


like this (if you can read this teeny print):


or even easier, using trig identites:


Then, you can get the final answer from this point:


by substituting for all four exponentials, and then simplifying (I'm not sure I did all of this right):
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September 12th, 2010, 06:01 PM   #6
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Re: Express in a+bi form

I posted too soon. There is an error in that last post. You can't expand using Euler's formula.

This is the whole problem with, I think, no mistakes:

Define:


Evaluate:


Use Euler's formula:



Solution (I think):


That's as simplified as I can get it. There is no imaginary component.
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October 6th, 2010, 07:18 PM   #7
 
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Re: Express in a+bi form

let c = a + i*b where a and b are real.

then



Giving us the more aesthetically pleasing answer (in z = Re(z) + i*Im(z) form):



The cosh and sinh are hyperbolic functions, which are defined by the exponential substitutions I utilized to get to the last step. http://en.wikipedia.org/wiki/Hyperbolic_function
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October 7th, 2010, 07:06 AM   #8
 
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Re: Express in a+bi form

I just noticed I somehow forgot the b in the expansion (where a and b are real valued) , so wherever you see in the above it should be .

The final answer is thus supposed to be

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