I am working on this problem below.
Prove that ,
So, I tried to compute , which I think is equal to
But here I need to show that , and I'm stuck on showing this. Can anyone give some help?
This should point you in the right direction,
Gamma function - Wikipedia, the free encyclopedia
please let me know if you require any further assistance
Great problem!
David
Thanks for your help David.
I already showed that
by a direct computation using the fact that . So, just take the absolute value of both sides then I got the result.
However,I'm still stuck with proving my original question. From the link you gave me, there are lots of ways to define the gamma function, so I think I should use the standard definition
Then I want to show ,
I have
How do I go from here, anyone can give a hand is appreciated.
Ahh you obviously didn't read the post in too much detail !! (j/j), there is a fantastic property of the gamma function in terms evaluating the conjugate of F(z) Whereby
Conjugate(F(z)) = F(Conjugate of z)
(I can prove if you want (more than happy to) but its a lot of work...)
Hope this points you in the right direction, if you need any further help please message me - this is a great problem!!
Regards,
David
NamelessGuy,
I apologise if I caused any offence, It was not my intention to do so, I only commented on the 'Conjugate' property as I thought you may not have read it.
Please let me know how you got the solution out, I am highly interested in comparing methodologies.
Regards,
David
Nope, you didn't offense me at all. I'm here to learn and get help so any help is welcome and appreciate.
If you're asking about the solution I got for this. Yes, I did but I didn't prove the conjugation property of the gamma function.
This is how I did
We know that
Hence,
Now compute
(here I skipped the step where I evaluated and by using Euler's formula.
So , and we obtain the result by taking square root.