Let be an odd function, integrable everywhere, with
Let be an even function, integrable everywhere, with
Prove the following:
- for all ;
- ;
- .
- Proof. We compute using the given properties,
- Again, we compute using the given properties of and ,
- Finally,
a) g(x) = f(x + 5) => given
g(-x) = f(-x + 5) =>
g(x) = f( – [x – 5] ) => g(x) is even
g(x) = -f(x – 5) => f(x) is odd
-g(x) = f(x – 5)
I think the book says that f is odd and g is even, so there is a mistake above.
in b , after the third = sign shouldnt there be a minus sign ?
He did two steps in one there.
Solutions here a) and b) are wrong, because he used that g is odd (and not even, as is asked in the book).
But the correct proofs are very similar to what he did, you just need to use integral properties in a) as well.