Mathematical Tools for Physics

2—Infinite Series 46

If I multiply this byxI get 2 x+ 3x^2 + 4x^3 + 5x^4 +···and that starts to look like a derivative.

xf(x) = 2x+ 3x^2 + 4x^3 + 5x^4 +···=

d

dx

(

x^2 +x^3 +x^4 +···

)

Again, the geometric series pops up, though missing a couple of terms.

xf(x) =

d

dx

(

1 +x+x^2 +x^3 +···− 1 −x

)

=

d

dx

[

1

1 −x

− 1 −x

]

=

1

(1−x)^2

− 1

The final result is then

f(x) =

1

x

[

1 −(1−x)^2

(1−x)^2

]

=

2 −x

(1−x)^2

2.8 Diffraction
When light passes through a very small opening it will be diffracted so that it will spread out in a characteristic
pattern of higher and lower intensity. The analysis of the result uses many of the tools that you’ve looked at in
the first two chapters, so it’s worth showing the derivation first.
The light that is coming from the left side of the figure has a wavelengthλand wave numberk= 2π/λ.
The light passes through a narrow slit of width=a. The Huygens construction for the light that comes through
the slit says that you can effectively treat each little part of the slit as if it is a source of part of the wave that
comes through to the right. (As a historical note, the mathematical justification for this procedure didn’t come
until about 150 years after Huygens proposed it, so if you think it isn’t obvious why it works, you’re right.)

y

dy

ysinθ

θ

r

r 0