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Trig Fresnel Integral
Asked 9 years ago
Modified 6 years, 2 months ago
Viewed 2k times
8
I'm confused with this integral because the square is on the x, not the whole function. How
can I integrate it? Thank you.
3
I have not done complex analysis (only real analysis as I am a high school student) so how
can I evaluate it using elementary functions (without complex analysis)?
real-analysis
integration
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definite-integrals
fresnel-integrals
edited Feb 10, 2016 at 22:14
user228113
asked May 1, 2013 at 8:35
please delete me
1,536
I'm not sure, would
be correct? – Niklas R May 1, 2013 at 8:38
2
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12
Let
Now, the Gamma function comes in handy:
But, since
Let
, we have:
:
This is a rather famous integral and can be found here and there. I leave it's evaluation to
the reader.
But, it evaluates to
, and so does the Fresnel in question.
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answered May 3, 2013 at 21:59
Cody
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I'm not completely following how you get equation 3 from equation 2. Can you help me out? – Bitrex
May 3, 2013 at 23:04
It comes from the Gamma function. You can use this as a parameter in the integral.
– Cody May 4, 2013 at 11:24
I follow now. Thanks! – Bitrex May 4, 2013 at 15:25
An nice integration, if you're interested, is to derive
. You can do it with
Laplace transforms instead of the classic residue method. – Cody May 5, 2013 at 13:33
The integral does not have an elementary antiderivative. If you want to avoid complex
3
analysis, try this method.
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answered May 2, 2013 at 2:51
Potato
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Your link is dead – user5389726598465 Jun 30, 2017 at 1:26
@user135711 Archive copy here: web.archive.org/web/20120524131816/http://… – Potato Jun 30,
2017 at 19:08