Complex Integration

  • Thread starter Yegor
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  • #1
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Main Question or Discussion Point

[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]???
 

Answers and Replies

  • #2
mathman
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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.
 
  • #3
lurflurf
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Yegor said:
[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]???
[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:
  • #4
lurflurf
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mathman said:
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.
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.
 

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