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Difficult integral

  1. Jul 22, 2009 #1
    How can I calculate the integral of sin(x^2) from zero to infinity? Do I need to use the residue theorem?
    I need a detailed answer, because i'm very new in this subject.
    I don't even know where to start.
    Thanks for any kind of help :-)
  2. jcsd
  3. Jul 22, 2009 #2


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    I solved it, but I don't know what your level of calculus is so if this is too hard.

    First I extended the integral to -infinity to infinity and wrote it as*
    [tex]\int \sin(x^2) \, \mathrm dx = \operatorname{Im} \int e^{i x^2} \, \mathrm dx[/tex].
    I happen to know that
    [tex]\int e^{- a x^2} \, \mathrm dx = \sqrt{ \frac{\pi}{a} }[/tex]
    if Re(a) > 0, and that the limit Re(a) -> 0 is well-defined.

    Combining all that allowed me to solve the integral, eventually I got
    [tex]\frac12 \sqrt{\frac{\pi}{2}}[/tex].

    * All integrations are over [itex]]-\infty, \infty[[/itex] unless otherwise specified
  4. Jul 22, 2009 #3
    Okay, thanks.
    I've got 2 questions: Any chance I can solve this problom with the residue theorem?
    I know that i can write sin(x^2) like this:x^2 - x^6/3! + x^10/5! - x^14/7!...
    But then what?
    And i can't see clearly enough the text you wrote. Do I need some kind of program in order to view your solution?
  5. Jul 22, 2009 #4


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    Yes you can use the residue theorem to solve this integral. Use the contour shown in http://en.wikipedia.org/wiki/Fresnel_integral#Error_function.

    Hint: start with the function [itex]e^{iz^2}[/itex] and evaluate it along the different parts of the contour.

    [quote='Gina88]And i can't see clearly enough the text you wrote. Do I need some kind of program in order to view your solution? [/quote]

    Are you using internet explorer by any chance, perhaps an outdated version? If so installing Firefox should solve the display issues.
    Last edited: Jul 22, 2009
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