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Integrating with Gamma: cos(theta)^(2k+1)
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In summary, you can evaluate the definite integral of an arbitrary odd-power of cosine using a reducing formula, and then relate that to the factorial form of the gamma function.
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Simon Bridge
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You mean:
[tex]\int_0^{\frac{\pi}{2}} \cos^{2k+1}(\theta)d\theta[/tex]... eg: evaluate the definite integral of an arbitrary odd-power of cosine.
The standard approach is to start by integrating by parts.
You'll end up with a reducing formula which you can turn into a ratio of factorials - apply the limits - after which it is a matter of relating that to the factorial form of the gamma function.
eg. http://mathworld.wolfram.com/CosineIntegral.html
[tex]\int_0^{\frac{\pi}{2}} \cos^{2k+1}(\theta)d\theta[/tex]... eg: evaluate the definite integral of an arbitrary odd-power of cosine.
The standard approach is to start by integrating by parts.
You'll end up with a reducing formula which you can turn into a ratio of factorials - apply the limits - after which it is a matter of relating that to the factorial form of the gamma function.
eg. http://mathworld.wolfram.com/CosineIntegral.html
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Simon Bridge
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If it was easy there'd be no point setting it as a problem.
I'm not going to do it for you ...
Do you know what a gamma function is? You can represent it as a factorial?
Can you identify where you are having trouble seeing what is going on?
Perhaps you should try to do the derivation for yourself?
I'm not going to do it for you ...
Do you know what a gamma function is? You can represent it as a factorial?
Can you identify where you are having trouble seeing what is going on?
Perhaps you should try to do the derivation for yourself?
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Simon Bridge
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Really? And I thought I was being mean...
The trig-form of the beta function aye - yep, that's a tad more elegant that the path I was suggesting before (the more usual one)... but relies on a hand-wave: do you know how the beta function is derived?
Also - you have [itex]\frac{1}{2}B(\frac{1}{2},k+1)[/itex] but you've spotted that.
If you look at the cosine formula - you have to evaluate the limits ... at first it looks grim because it gives you a sum of terms like [itex]\sin\theta\cos^{2k}\theta[/itex] which is zero at both limits ... unless k=0 ... which is the first term in the sum, which is 1.
After that it is a matter of subbing in the factorial representation of the gamma function.
Which would be a concrete proof.
Yours is shorter and if you have the beta function in class notes then you should be fine using it.
The trig-form of the beta function aye - yep, that's a tad more elegant that the path I was suggesting before (the more usual one)... but relies on a hand-wave: do you know how the beta function is derived?
Also - you have [itex]\frac{1}{2}B(\frac{1}{2},k+1)[/itex] but you've spotted that.
If you look at the cosine formula - you have to evaluate the limits ... at first it looks grim because it gives you a sum of terms like [itex]\sin\theta\cos^{2k}\theta[/itex] which is zero at both limits ... unless k=0 ... which is the first term in the sum, which is 1.
After that it is a matter of subbing in the factorial representation of the gamma function.
Which would be a concrete proof.
Yours is shorter and if you have the beta function in class notes then you should be fine using it.
FAQ: Integrating with Gamma: cos(theta)^(2k+1)
What is the concept of integrating with Gamma?
Integrating with Gamma involves using the Gamma function, which is an extension of the factorial function, to solve integrals involving trigonometric functions such as cos(theta)^(2k+1). It allows for the evaluation of integrals that cannot be solved using traditional methods.
What is the formula for the Gamma function?
The Gamma function is defined as Γ(z) = ∫0∞ x^(z-1)e^(-x) dx. This can also be written as Γ(z) = (z-1)!, where z is a complex number.
How is the Gamma function used in integration with cos(theta)^(2k+1)?
The Gamma function can be used to rewrite cos(theta)^(2k+1) as a power of a sine function, which can then be integrated using substitution or integration by parts. This technique is particularly useful for integrals involving odd powers of trigonometric functions.
What are the benefits of using the Gamma function in integration?
The Gamma function allows for the evaluation of integrals that would otherwise be difficult or impossible to solve using traditional methods. It also has many applications in statistics, probability, and other areas of mathematics.
Are there any limitations or drawbacks to using the Gamma function in integration?
While the Gamma function is a powerful tool for solving integrals involving cos(theta)^(2k+1), it may not always be the most efficient method. It is important to consider the complexity of the integral and whether other techniques may be more suitable. Additionally, the Gamma function may not be well-defined for certain values of z, so caution must be taken when applying it in integration problems.