Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex Integration

  1. Jul 4, 2005 #1
    [tex]\int_0^\infty e^{-ax^2+bx} dx[/tex], a and b may be complex.
    Does exist any formula for this integral?
    Or for [tex]\int_{-\infty}^\infty e^{-ax^2+bx} dx[/tex]???
     
  2. jcsd
  3. Jul 4, 2005 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    ax2-bx=a(x-b/2a)2+(b/2)2/a

    Your first integral then becomes an erf integral, which can only be done numerically. The second integral has a closed form solution - integral of Gaussian.
     
  4. Jul 5, 2005 #3

    lurflurf

    User Avatar
    Homework Helper

    [tex]\int_{-\infty}^\infty e^{-ax^2} dx={\sqrt{\frac{\pi}{a}}}}[/tex]
    Consider your integral multiply it by a constant of the form exp(c) where c lets you conplete the square of the quadratic. Then observe
    [tex]\int_{-\infty}^\infty e^{-x^2} dx=\int_{-\infty}^\infty e^{-(x+y)^2} dx
    [/tex]
    for any constant y
     
    Last edited: Jul 5, 2005
  5. Jul 5, 2005 #4

    lurflurf

    User Avatar
    Homework Helper

    It is true that the first will involve erf, while the second will have nicer form. Yet erf is a closed form. Also closed form verses numerical solution is kind of silly any way. log(2) is a closed form, but if you want a number you have to "do it numerically". The issue has more do do with how many function one want to define tabulate and use. The distinction between an answer erf(1) and one of sin(1) is mostly historical.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Complex Integration
  1. Complex Integral (Replies: 5)

  2. Complex integral (Replies: 16)

  3. Complex Integral (Replies: 6)

  4. A complex integral (Replies: 2)

  5. Complex integral (Replies: 3)

Loading...