Mathematical Tools for Physics

14—Complex Variables 434

depend on the relative sign ofaandb, because the change of variablesθ′=θ+πchanges the coefficient ofb
while the periodicity of the cosine means that you can leave the limits alone. I may as well assume thataandb
are positive. The trick now is to use Euler’s formula and express the cosine in terms of exponentials.

Letz=eiθ, then cosθ=

1

2

[

z+

1

z

]

and dz=ieiθdθ=iz dθ

Asθgoes from 0 to 2 π, the complex variablezgoes around the unit circle. The integral is then

∫ 2 π

0

dθ

1

(a+bcosθ)

=

∫

C

dz

iz

1

a+b

(

z+^1 z

)

/ 2

The integrand obviously has some poles, so I have to locate them.

2 az+bz^2 +b= 0 has roots z=

− 2 a±

√

(2a)^2 − 4 b^2

2 b

=z±

Becausea > b, the roots are real. The important question is: Are they inside or outside the unit circle? The
roots depend on the ratioa/b=λ.

z±=

[

−λ±

√

λ^2 − 1

]

(14)

Asλvaries from 1 to∞, the two roots travel from− 1 → −∞and from− 1 → 0 , soz+stays inside the unit
circle (problem 19 ). The integral is then

−

2 i

b

∫

C

dz

z^2 + 2λz+ 1

=−

2 i

b

∫

C

dz

(z−z+)(z−z−)

=−

2 i

b

2 πiRes

z=z+

=−

2 i

b

2 πi

1

z+−z−

=

2 π

b

√

λ^2 − 1

=

2 π

√

a^2 −b^2