Mathematical Tools for Physics

2—Infinite Series 63

2.38 A function is defined by the integral

f(x) =

∫x

0

dt

1 −t^2

Expand the integrand with the binomial expansion and derive the power (Taylor) series representation forfabout
x= 0. Also make the hyperbolic substitution to evaluate it in closed form.

R q

p

q

2.39 Light travels from a point on the right (p), hits a spherically shaped
mirror and goes to a point (q). The radius of the spherical surface isRand
you can parametrize the point on the surface by the angleθfrom the center
of the sphere. Compute the time it takes light to travel on the indicated path
(two straight line segments) as a function of the angleθ.
Expand the time through second order in a power series inθand show that
the functionT(θ)has a minimum if the distanceqis small enough, but that
it switches to a maximum whenqexceeds a particular value.
This is the focus.

2.40 The quadratic equationax^2 +bx+c= 0is almost a linear equation ifais small enough:bx+c= 0⇒
x=−c/b. You can get a solution iteratively by rewriting the equation as

x=−

c

b

−

a

b

x^2

Solve this by neglecting the second term, then with this approximate value ofxget an improved value of the root
by

x 2 =−

c

b

−

a

b

x^21

and you can repeat the process. For comparison take the exact solution and do a power series expansion on it
for smalla. See if the results agree.
Where does the other root come from? Thatxis very large, so the first two terms in the quadratic are the big
ones and must nearly cancel.ax^2 +bx= 0sox=−b/a. Rearrange the equation so that you can iterate it.

x=−

b

a

−

c

ax