Is int(F*(x-y/2)F(x-y/2))dy always a negative value?

[pivoxa15]

Take int(F*(x-y/2)F(x-y/2))dy over all y. Where F is a complex function

Let A=x-y/2

dA/dy=-1/2 => dy=-2dA

So -2int(F*(A)F(A))dA = int(F*(x-y/2)F(x-y/2))dy

but int(F*(A)F(A))dA >0 so is int(F*(x-y/2)F(x-y/2))dy but from the result it looks as if int(F*(x-y/2)F(x-y/2))dy<0

What is going on?

 

[neutrino]

Isn't there a -2 somewhere in the new integral?

 

[HallsofIvy]

Take int(F*(x-y/2)F(x-y/2))dy over all y. Where F is a complex function

Let A=x-y/2

dA/dy=-1/2 => dy=-2dA

So -2int(F*(A)F(A))dA = int(F*(x-y/2)F(x-y/2))dy

but int(F*(A)F(A))dA >0 so is int(F*(x-y/2)F(x-y/2))dy but from the result it looks as if int(F*(x-y/2)F(x-y/2))dy<0

What is going on?

What happened to your limits of integration? Replacing a function of y by a function of A will "reverse" the limits of integration.

 

[pivoxa15]

Right. Good one.