Advanced book on Mathematics Olympiad

3.3 Multivariable Differential and Integral Calculus 167

∫ 1

0

e^2 πimxdx=

1

2 πim

e^2 πimx

∣

∣∣

∣

1

0

= 0.

Therefore, equality holds term by term for the Fourier series. The theorem is proved.

If after this example you don’t love Fourier series, you never will. Below are listed

more applications of the Fourier series expansion.

493.Prove that for every 0<x< 2 πthe following formula is valid:

π−x

2

=

sinx

1

+

sin 2x

2

+

sin 3x

3

+···.

Derive the formula

π

4

=

∑∞

k= 1

sin( 2 k− 1 )x

2 k− 1

,x∈( 0 ,π).

494.Use the Fourier series of the function of period 1 defined byf(x)=^12 −xfor

0 ≤x<1 to prove Euler’s formula

π^2

6

= 1 +

1

22

+

1

32

+

1

42

+···.

495.Prove that

π^2

8

= 1 +

1

32

+

1

52

+

1

72

+···.

496.For a positive integernfind the Fourier series of the function

f(x)=

sin^2 nx

sin^2 x

497.Letf :[ 0 ,π]→Rbe aC∞function such that(− 1 )nf(^2 n)(x) ≥ 0 for any

x∈[ 0 ,π]andf(^2 n)( 0 )=f(^2 n)(π )=0 for anyn≥0. Show thatf(x)=asinx

for somea>0.

3.3 Multivariable Differential and Integral Calculus.....................

3.3.1 Partial Derivatives and Their Applications....................

This section and the two that follow cover differential and integral calculus in two and
three dimensions. Most of the ideas generalize easily to then-dimensional situation.
All functions below are assumed to be differentiable. For a two-variable function this