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Complex integration

281
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How do I solve an integral of the type

[tex]\int f(v) e^{iavx} dv[/tex] ?

Can I just treat i as any other constant?
 

Answers and Replies

554
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I'm not entirely sure I am correct about this but it seems logical to expand the complex exponent and integrate it further from there.
[tex]e^{i \phi} = \cos (\phi) + i \sin (\phi)[/tex]

At a guess I would say yes, [tex]i[/tex] is a constant... Just a logical guess though...
 
281
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[tex]\int f(v) e^{iavx} dv = \int f(v) \left( \cos{avx} + i \sin{avx} \right) dv =[/tex]
[tex]= \int f(v) \cos{avx} dv + i \int f(v) \sin{avx} dv[/tex]

Maybe?
 
Yes, you can treat [itex]i[/itex] as a constant.

Or you can use Euler's formula and write it as the sum of cos and sin, yes.
 
554
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Using Euler's formula doesn't get rid of the [tex]i[/tex] ofcourse...

Logarythmic, looks fine by me as long as you put [tex]avx[/tex] in brackets ;)
 
281
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Got it, so
[tex]w(x) = \int_{-u_0}^{u_0} i2 \pi v e^{i2 \pi vx} dv = \frac{1}{\pi x^2} \left[ 2 \pi u_0 x \cos{(2 \pi u_0 x)} - \sin{(2 \pi u_0 x)} \right][/tex]
 
554
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I got the same, so I guess it's correct.
 

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