Hey, i am little puzzled by this question.... i get that none of the sets (a,b or c) are orthogonal when taken over -infinity to infinity.

 

am i doing something wrong, i would have thought at least one of them would be orthogonal...???

 

this is the problem in question:

 

the inner product i think i am meant to be using is: <f|g> = ?f*g

 

where the thing on the left (f*) is complex conjugate of f.

 

 

the combinations i tried were:

 

(a) cos(x) & 3sin(x)

(b) 1 & (-x+1)

© x * 4x^{3}

 

i would have thought that cos(x) & 3sin(x) would be orthogonal...?? or is it that i am not meant to be integrating from -infinty to infinity? (that is should i just be going from 0 to 2pi?)

 

-a puzzled Sarah :S

ok i have been told that they none are othorgoanl sets. but i know that (-x+1) & (x^2 - 4x +2)/2 is orthogonal. but i am can't seem to show it.... i just get inifinity terms...

i would have thought that cos(x) & 3sin(x) would be orthogonal...?? or is it that i am not meant to be integrating from -infinty to infinity? (that is should i just be going from 0 to 2pi?)

 

The inner product for those two functions on [imath](-\infty' date='\infty)[/imath'] isn't even defined, because the limit you obtain when you evaluate the resulting integral doesn't exist. Is this question from the same instructor as the Wronskian question?

 

but i know that (-x+1) & (x^2 - 4x +2)/2 is orthogonal. but i am can't seem to show it.... i just get inifinity terms...

 

Well then they aren't orthogonal! The inner product of two orthogonal functions is zero. I wouldn't expect them to be orthogonal anyway because their product has mixed symmetry (or no symmetry, if you prefer).

phew, ok.

 

so are their any pairs in these sets (other than cos and sin if evaluated between 0 and 2pi) which are orthogonal, because i don't think there is, i just want to make sure i am not doing somethign wrong here

None of these sets are orthogonal.

 

Set (a): The inner product isn't defined on [imath](-\infty,\infty)[/imath] for any pair of these functions.

 

Set (b): All of the functions in this set have mixed symmetry (except the first one, which has even symmetry).

 

Set ©: The functions [imath]f(x)=x[/imath] and [imath]f(x)=4x^3[/imath] are both orthogonal to [imath]f(x)=e^{-ax^2}[/imath] on [imath](-\infty,\infty)[/imath], but they aren't orthogonal to each other.

ok cool, thats what i was hoping for!

 

Thank you very much Tom!