• yungman
In summary: But the negative is more difficult because it is not always clear what the negative sign means. If the function is cosine, the negative sign means that the function goes down instead of up. If you are trying to find the series representation of a function with a negative argument, you might need to consult a calculus textbook or look at examples online.I am studying on my own, this question only on the first section, before a lot of the information your link shows. I have to spend some time looking at this.
yungman
Can you show me how to get the series representation of $$\Gamma$$(n-3/2+1)?

For example $$\Gamma$$(n+3/2+1)=$$\frac{(2n+3)(2n+1)!}{2^{2n+2}.n!}$$.

I cannot figure out how to write a series with:

n=0 => $$\Gamma$$(0-3/2+1)= -2$$\sqrt{\pi}$$

n=1 => $$\Gamma$$(1-3/2+1)= $$\sqrt{\pi}$$

n=2 => $$\Gamma$$(2-3/2+1)= 1/2$$\sqrt{\pi}$$

n=3 => $$\Gamma$$(3-3/2+1)= 4/3$$\sqrt{\pi}$$

This is not homework. I have spent 2 days on this and can't figure it out!

Thanks

Alan

If I understand you correctly, the trick is to replace n by n-3:
$$\Gamma(n - \frac 3 2 +1) = \Gamma((n-3) + \frac 3 2 + 1) =\frac{(2(n-3)+3)(2(n-3)+1)!}{2^{2(n-3)+2}(n-3)!}=\frac{(2n-3)(2n-5)!}{2^{2n-4}(n-3)!}$$

At least I think this works for $n \ge 3$.

LCKurtz said:
If I understand you correctly, the trick is to replace n by n-3:
$$\Gamma(n - \frac 3 2 +1) = \Gamma((n-3) + \frac 3 2 + 1) =\frac{(2(n-3)+3)(2(n-3)+1)!}{2^{2(n-3)+2}(n-3)!}=\frac{(2n-3)(2n-5)!}{2^{2n-4}(n-3)!}$$

At least I think this works for $n \ge 3$.

Thanks for trying. But that was where I got really stuck. I can easily for find the series representation for n=1,2,3,4... It is the n=0 that is the problem. The series solution has to cover n=0,1,2...

Actually I am working on the problem is to show

J-3/2(x)=$$\sqrt{\frac{2}{\pi x}}$$[$$\frac{-cos(x)}{x}$$-sin(x)]

Where this is solution of Bessel function.

Please, even if you have any suggestion, I would like to listen to it. I am very despirate! I absolutely ran out of ideas!

LCKurtz said:
It has been many years since I looked at Bessel functions, so I can't help you more. I assume you have already looked at resources on the web such as this wikipedia article:

http://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind_:_J.CE.B1

If you haven't already looked there you might find something useful. Good luck.

Thanks

I am studying on my own, this question only on the first section, before a lot of the information your link shows. I have to spend some time looking at this. Yes I did look at this before.

The question in the book ask both J(3/2) and J(-3/2). The possitive is relative easy because the gamma function always stay possitive.

## 1. What is the Gamma function?

The Gamma function is a mathematical function that is defined for all complex numbers except negative integers. It is denoted by Γ(z) and is an extension of the factorial function for non-integer values.

## 2. How is the Gamma function useful?

The Gamma function has various applications in mathematics, physics, and engineering. It is used in the evaluation of definite integrals, solving differential equations, and in probability and statistics. It also has connections to other important mathematical functions such as the Beta function and the Riemann zeta function.

## 3. What is the series representation of the Gamma function?

The series representation of the Gamma function is given by Γ(z) = ∫0 tz-1e-tdt = 1/z - γ + Σn=1 (-1)n z(z+1)...(z+n-1)/n! for Re(z) > 0, where γ is the Euler-Mascheroni constant.

## 4. How do you convert the Gamma function to a series?

To convert the Gamma function to a series, we can use the series representation mentioned above. First, we express the Gamma function as an integral and then use a series expansion for the integrand. We can then rearrange the terms and simplify to obtain the series representation of the Gamma function.

## 5. Are there any other methods for computing the Gamma function?

Yes, there are other methods for computing the Gamma function, such as the Lanczos approximation and the Stirling's formula. These methods are often more efficient and accurate for large values of z. There are also various software libraries and packages available for computing the Gamma function in different programming languages.

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