Proof of Parseval's Identity for a Fourier Sine/Cosine transform

[tanaygupta2000]

Can anyone help me with the Proof of Parseval Identity for Fourier Sine/Cosine transform :

2/π [integration 0 to ∞] Fs(s)•Gs(s) ds = [integration 0 to ∞] f(x)•g(x) dx

I've successfully proved the Parseval Identity for Complex Fourier Transform, but I'm unable to figure out from where does the term '2/π' comes in the Parseval formula for Fourier Sine and Cosine Transform.
Any help will be appreciated.
Thank You.

 

[mathman]

It would help if you defined your terms and variables.

 

[tanaygupta2000]

I'm having difficulty in proving this. According to me, the term '2/π' is not coming.

 

[mathman]

It would help if the definitions were given for ##F_C(s),G_C(s)##, etc.

 

[tanaygupta2000]

Refer this picture

 

[mathman]

I haven't tried to work it through, but it looks like the factor ##\sqrt{\frac{2}{\pi}}## appearing twice in setting up the integral leads to ##\frac{2}{\pi}##.

 

[tanaygupta2000]

I've proved it the same way as Complex transform identity.

 

[mathman]

I've proved it the same way as Complex transform identity.View attachment 239848

Please post attachment horizontally.

 

[tanaygupta2000]

Help proving this :

 

[tanaygupta2000]

Help proving this :View attachment 239908

Correct formula is this :

 

[mathman]

http://www.math.pitt.edu/~troy/stochastic/parseval.pdf

The above may help.

 

[tanaygupta2000]

I've already proved Complex Identity as given in the link. What I have trouble in proving is Sine/Cosine Parseval Identity (which is not there in the link)

http://www.math.pitt.edu/~troy/stochastic/parseval.pdf

The above may help.

 

[mathman]

Can't you simply separate the real and imaginary parts of the complex identity?