# Orthogonal functions

1. Nov 12, 2008

### dirk_mec1

1. The problem statement, all variables and given/known data

http://img55.imageshack.us/img55/8494/67023925dy7.png [Broken]

3. The attempt at a solution

All functions orthogonal to 1 result in the fact that: $$\int_a^b f(t)\ \mbox{d}t =0$$

Now the extra condition is that f must be continous. (because of the intersection).

But where does the fact that f(a)=f(b)=0 comes from? And why look at the deratives?

Last edited by a moderator: May 3, 2017
2. Nov 12, 2008

### Dick

Remember way back in first year calc, when you learned that to do that integral you find an antiderivative F(x) and evaluate F(b)-F(a). This is that same problem in disguise.

3. Nov 12, 2008

### dirk_mec1

Well I thought of this: $$\int_a^b \int_a^t f(s)\ \mbox{d}s \mbox{d}t =0$$

4. Nov 12, 2008

### Dick

Fine. What are you going to do with it? Why don't you just define $$F(x)=\int_a^x f(s)\ \mbox{d}s$$
What are some of the properties of F(x)?