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August 17th, 2012, 11:54 AM   #1
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Evaluation of (sin x)^m (cos x)^n dx

Given the values of m & n how would you evaluate

without means of Gamma function?
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August 17th, 2012, 12:49 PM   #2
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Re: Evaluation of (sin x)^m (cos x)^n dx

Just a Hint

I would first rewrite is like this



using integrating by parts







I would continue until I get sin(x)
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August 17th, 2012, 02:41 PM   #3
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Re: Evaluation of (sin x)^m (cos x)^n dx

Hello, Rejjy!

Quote:











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[color=beige]. . [/color]







~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~








[color=beige]. . . . . . [/color]

[color=beige]. . . . . . [/color]

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August 18th, 2012, 02:19 AM   #4
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Re: Evaluation of (sin x)^m (cos x)^n dx

What about something along the lines of m=7/2 and n=3/2.
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August 18th, 2012, 08:46 AM   #5
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Re: Evaluation of (sin x)^m (cos x)^n dx

And what if m and n are very large?
I have a book that states the following result.

Also, it says to multiple the above by if both m and n are even.

@The_Fool
Agreed, that is an interesting question. However, I am not aware if the above result holds for fractional m & n. It seems like it won't.

Regards,
Rejjy
18-Aug-2012
21:16 IST
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August 18th, 2012, 10:10 AM   #6
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Re: Evaluation of (sin x)^m (cos x)^n dx

Quote:
Originally Posted by The_Fool
What about something along the lines of m=7/2 and n=3/2.
If my calculations are correct (I'm pretty sure they are correct) :



where is the complete elliptic integral of the first kind and is the gamma function.

Balarka

.
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August 18th, 2012, 10:22 AM   #7
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Re: Evaluation of (sin x)^m (cos x)^n dx

Now for the original problem, I would prefer zaidalyafey's method :

if then,





where which is elementary and easy to calculate.

Balarka

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August 18th, 2012, 03:24 PM   #8
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Re: Evaluation of (sin x)^m (cos x)^n dx

Quote:
Originally Posted by mathbalarka
Quote:
Originally Posted by The_Fool
What about something along the lines of m=7/2 and n=3/2.
If my calculations are correct (I'm pretty sure they are correct) :



where is the complete elliptic integral of the first kind and is the gamma function.

Balarka

.
But, you used the gamma function. The original post asked for a way to evaluate it without the gamma function.
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August 18th, 2012, 11:22 PM   #9
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Re: Evaluation of (sin x)^m (cos x)^n dx

Quote:
Originally Posted by The_Fool
But, you used the gamma function. The original post asked for a way to evaluate it without the gamma function.
It's impossible to do it without any type of special function for
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August 19th, 2012, 02:16 AM   #10
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Re: Evaluation of (sin x)^m (cos x)^n dx

Well, I don't think that's right anyway since that integral is real:

I used the beta function instead of the gamma function. While beta can be defined by gamma, it can stand on it's own without it in several other forms.
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