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 Post subject: Posted: Sat, 23 Feb 2008 20:06:11 UTC
 Senior Member

Joined: Thu, 26 Jul 2007 19:08:57 UTC
Posts: 61
Find :

Last edited by M.A on Fri, 14 Mar 2008 19:16:57 UTC, edited 3 times in total.

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 Post subject: Posted: Sun, 24 Feb 2008 05:58:39 UTC
 Senior Member

Joined: Mon, 12 Nov 2007 00:25:37 UTC
Posts: 106
¬logical wrote:

Are the denominators of the 2nd and 3rd expression purposely equal or is it a mistake?

i think that is on purpose.

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 Post subject: Posted: Sun, 24 Feb 2008 06:00:10 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15563
Location: Austin, TX
¬logical wrote:

Are the denominators of the 2nd and 3rd expression purposely equal or is it a mistake?

i think that is on purpose.

Why? Usually such things have an interesting sort of symmetry for a reason.

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 Post subject: Posted: Mon, 25 Feb 2008 14:01:51 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
Prove that for

Spoiler:
This one is quite simple.
Let , then integrate by parts.
Let , and .
Therefore,
We obtain etc.

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Posted: Tue, 4 Mar 2008 22:00:23 UTC
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Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
Prove that .

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Posted: Tue, 4 Mar 2008 22:50:04 UTC
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Joined: Sat, 18 Mar 2006 08:42:24 UTC
Posts: 834
bidentate wrote:
Prove that .

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 Post subject: Posted: Wed, 2 Apr 2008 23:33:59 UTC
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Joined: Sat, 18 Mar 2006 08:42:24 UTC
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1) Using the auxiliary function prove that

2) Fresnel Integrals: Using the complex auxiliary function prove that:

Hint:
Spoiler:
Find and first!

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 Post subject: Posted: Thu, 15 May 2008 00:39:33 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
Evaluate the limit: .

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Posted: Mon, 19 May 2008 16:04:15 UTC
 S.O.S. Oldtimer

Joined: Thu, 13 Jul 2006 19:47:54 UTC
Posts: 308
bidentate wrote:
Evaluate the limit: .

Spoiler:

evaluated at y = 0.

From here it is simple to show that the answer is 2.

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 Post subject: Posted: Mon, 19 May 2008 20:40:44 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
Zathras wrote:
Spoiler:

evaluated at y = 0.

From here it is simple to show that the answer is 2.

Correct!

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Posted: Tue, 22 Jul 2008 20:21:50 UTC
 Member

Joined: Tue, 22 Jul 2008 09:18:49 UTC
Posts: 24
Location: Poltergeist
Hi.

commutative wrote:
1) Using the auxiliary function prove that

This one is a great way to prove the result ! ^^

Spoiler:
We first have to prove that the 2 integrals in f(x) (uniformly if I remember) converge. This part will be left to theoricists who like dominating

Change variable of the second : .

Thus the function f is constant.
We can write in particular

Therefore
Because this latter integral uniformly converges, we can invert the integral and the limit, which is 0.

is a positive function, for any t. So its integral over is positive.

We can conclude that

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 Post subject: Posted: Wed, 15 Oct 2008 23:57:18 UTC
 Senior Member

Joined: Thu, 11 Sep 2008 01:59:14 UTC
Posts: 73
Here's a hard one... it was asked 10 yrs ago on a exam.

I can't even evaluate the borders becuz it gives an undetermined form and they'res a function in the result that seems very hard to derivate.

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