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#1 2012-11-01 07:46:06

zetafunc.
Guest

Fresnel Integrals via Differentiation Under Integral Sign

I've been trying to do

and

using DUIS, but I can't think of any kind of useful parametrisation that would work. Every time I do, I usually end up with something that *looks* like you can use integration by parts, but that doesn't work. I'm aware the indefinite integral form of these integrals can't be expressed in terms of elementary functions, so I'm hoping I might have more luck with the improper ones. Can anyone show me a useful parametrisation that would work here?

#2 2012-11-01 08:55:11

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,776

Re: Fresnel Integrals via Differentiation Under Integral Sign

Hi;

I have never seen it done using differentiation under the integral sign.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#3 2012-11-01 08:58:22

zetafunc.
Guest

Re: Fresnel Integrals via Differentiation Under Integral Sign

The last page here says you can do it.

tinyurl. com /bj99zrg

(remove spaces)

(I am assuming they meant cos(x[sup]2[/sup]) and not cos[sup]2[/sup]x because the latter is not a Fresnel integral and not very difficult.)

#4 2012-11-01 09:00:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,776

Re: Fresnel Integrals via Differentiation Under Integral Sign

Hi;

This page does it without resorting to DUIS.

http://mathforum.org/library/drmath/view/51877.html


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Online

#5 2012-11-01 09:02:18

zetafunc.
Guest

Re: Fresnel Integrals via Differentiation Under Integral Sign

I know you can do it without DUIS (e.g. gamma function method), but was curious if it was made really simple via DUIS (like integrating sinx/x).

#6 2012-11-01 09:05:54

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,776

Re: Fresnel Integrals via Differentiation Under Integral Sign

Hi;

(I am assuming they meant cos(x^2) and not cos^2x because the latter is not a Fresnel integral and not very difficult.)

If you meant the examples at the end of that pdf then it meant

and not the integral you want.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Online

#7 2012-11-01 09:07:06

zetafunc.
Guest

Re: Fresnel Integrals via Differentiation Under Integral Sign

But that is not a Fresnel integral. They said that the middle row contains two Fresnel integrals... so either they meant that it does not contain two Fresnel integrals, or they didn't parenthesise the powers properly...

#8 2012-11-01 09:12:25

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,776

Re: Fresnel Integrals via Differentiation Under Integral Sign

Those are not Fresnel integrals so he made one of two mistakes. He did not put the square in the proper spot or he does not know what a Fresnel Integral is.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Online

#9 2012-11-01 09:16:30

zetafunc.
Guest

Re: Fresnel Integrals via Differentiation Under Integral Sign

Hmm. So, there might not be a way. I will still try to look for one however. It seems I never get to use this little tool.

#10 2012-11-01 23:00:29

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,776

Re: Fresnel Integrals via Differentiation Under Integral Sign

Hi;

I tried a couple of parametrizations but did not have any luck producing one that gets the known answer.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Online

#11 2012-11-01 23:02:09

zetafunc.
Guest

Re: Fresnel Integrals via Differentiation Under Integral Sign

What sorts did you try? I tried something of the form ln(bsinx[sup]2[/sup]) or similar, in the hope that I could cancel the trig term, but it did not work. Maybe trying to use a trig identity would fare better?

#12 2012-11-01 23:10:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,776

Re: Fresnel Integrals via Differentiation Under Integral Sign

I tried that one and also sin(x^b).


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Online

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