Mixing
Problem proving the Fresnel integral
$begingroup$
I have shown that
$$int^infty_0 e^{ix^2}dx = dfrac{sqrtpi}{2},$$
using contour integration on $f(z) = e^{iz^2}$. But since
$$int^infty_0 e^{ix^2}dx = int^infty_0 (cos x^2+ i sin x^2) dx = dfrac{sqrtpi}{2}, $$
I would have thought equating the imaginary and real parts would do the trick. But this just gives wrong answers. What's wrong here?
complex-analysis fresnel-integrals
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
$endgroup$
add a comment |
$begingroup$
I have shown that
$$int^infty_0 e^{ix^2}dx = dfrac{sqrtpi}{2},$$
using contour integration on $f(z) = e^{iz^2}$. But since
$$int^infty_0 e^{ix^2}dx = int^infty_0 (cos x^2+ i sin x^2) dx = dfrac{sqrtpi}{2}, $$
I would have thought equating the imaginary and real parts would do the trick. But this just gives wrong answers. What's wrong here?
complex-analysis fresnel-integrals
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
$endgroup$
$begingroup$
Your first result is not correct. It should be $left(frac{1}{2}+frac{i}{2}right) sqrt{frac{pi }{2}}$
$endgroup$
– Claude Leibovici
Jan 16 at 11:08
$begingroup$
@ClaudeLeibovici, you're right. I made a dumb mistake. This fixes the problem. Thanks!
$endgroup$
– Stijn D'hondt
Jan 16 at 11:35
$begingroup$
You are very welcome !
$endgroup$
– Claude Leibovici
Jan 16 at 11:43
add a comment |
$begingroup$
I have shown that
$$int^infty_0 e^{ix^2}dx = dfrac{sqrtpi}{2},$$
using contour integration on $f(z) = e^{iz^2}$. But since
$$int^infty_0 e^{ix^2}dx = int^infty_0 (cos x^2+ i sin x^2) dx = dfrac{sqrtpi}{2}, $$
I would have thought equating the imaginary and real parts would do the trick. But this just gives wrong answers. What's wrong here?
complex-analysis fresnel-integrals
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
$endgroup$
I have shown that
$$int^infty_0 e^{ix^2}dx = dfrac{sqrtpi}{2},$$
using contour integration on $f(z) = e^{iz^2}$. But since
$$int^infty_0 e^{ix^2}dx = int^infty_0 (cos x^2+ i sin x^2) dx = dfrac{sqrtpi}{2}, $$
I would have thought equating the imaginary and real parts would do the trick. But this just gives wrong answers. What's wrong here?
complex-analysis fresnel-integrals
complex-analysis fresnel-integrals
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
asked Jan 16 at 10:35
Stijn D'hondtStijn D'hondt
62
62
$begingroup$
Your first result is not correct. It should be $left(frac{1}{2}+frac{i}{2}right) sqrt{frac{pi }{2}}$
$endgroup$
– Claude Leibovici
Jan 16 at 11:08
$begingroup$
@ClaudeLeibovici, you're right. I made a dumb mistake. This fixes the problem. Thanks!
$endgroup$
– Stijn D'hondt
Jan 16 at 11:35
$begingroup$
You are very welcome !
$endgroup$
– Claude Leibovici
Jan 16 at 11:43
add a comment |
$begingroup$
Your first result is not correct. It should be $left(frac{1}{2}+frac{i}{2}right) sqrt{frac{pi }{2}}$
$endgroup$
– Claude Leibovici
Jan 16 at 11:08
$begingroup$
@ClaudeLeibovici, you're right. I made a dumb mistake. This fixes the problem. Thanks!
$endgroup$
– Stijn D'hondt
Jan 16 at 11:35
$begingroup$
You are very welcome !
$endgroup$
– Claude Leibovici
Jan 16 at 11:43
$begingroup$
Your first result is not correct. It should be $left(frac{1}{2}+frac{i}{2}right) sqrt{frac{pi }{2}}$
$endgroup$
– Claude Leibovici
Jan 16 at 11:08
$begingroup$
Your first result is not correct. It should be $left(frac{1}{2}+frac{i}{2}right) sqrt{frac{pi }{2}}$
$endgroup$
– Claude Leibovici
Jan 16 at 11:08
$begingroup$
@ClaudeLeibovici, you're right. I made a dumb mistake. This fixes the problem. Thanks!
$endgroup$
– Stijn D'hondt
Jan 16 at 11:35
$begingroup$
@ClaudeLeibovici, you're right. I made a dumb mistake. This fixes the problem. Thanks!
$endgroup$
– Stijn D'hondt
Jan 16 at 11:35
$begingroup$
You are very welcome !
$endgroup$
– Claude Leibovici
Jan 16 at 11:43
$begingroup$
You are very welcome !
$endgroup$
– Claude Leibovici
Jan 16 at 11:43
add a comment |
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$begingroup$
Your first result is not correct. It should be $left(frac{1}{2}+frac{i}{2}right) sqrt{frac{pi }{2}}$
$endgroup$
– Claude Leibovici
Jan 16 at 11:08
$begingroup$
@ClaudeLeibovici, you're right. I made a dumb mistake. This fixes the problem. Thanks!
$endgroup$
– Stijn D'hondt
Jan 16 at 11:35
$begingroup$
You are very welcome !
$endgroup$
– Claude Leibovici
Jan 16 at 11:43