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Complex integration

  1. Apr 25, 2008 #1
    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?
  2. jcsd
  3. Apr 25, 2008 #2
    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...
  4. Apr 25, 2008 #3
    [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]

  5. Apr 25, 2008 #4
    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.
  6. Apr 25, 2008 #5
    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 ;)
  7. Apr 25, 2008 #6
    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]
  8. Apr 25, 2008 #7
    I got the same, so I guess it's correct.
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