Expanding f(x) in Legendre Polynomials: Applying the Transformation u = x/2

[Logarythmic]

I wish to expand

$$f(x) = 1 - \frac{x^2}{4} , -2 \leq x \leq 2$$

in terms of Legendre polynomials.

I know that the transformation

$$u = \frac{x}{2}$$

maps the function onto the interval (-1,1), but how do I apply this transformation?

Should I use

$$g(x) = uf(x)$$

or maybe

$$g(x) = f(u(x))$$

?

 

[HallsofIvy]

g(x)= f(u(x)). You are basically making a "change of variable".

 

[Logarythmic]

But shouldn't $$f(2) = g(1)$$?