Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
Taking the limits R !1and " ! 0
Z 1
0
sin x
x
dx ˆ
2 :
Problems
6.1. Given three complex numbers z 1 ˆ a ‡ ib, z 2 ˆ c ‡ id, and z 3 ˆ g ‡ ih,
show that:
(a) z 1 ‡ z 2 ˆ z 2 ‡ z 1
commutative law of addition;
(b) z 1 ‡…z 2 ‡ z 3 †ˆ…z 1 ‡ z 2 †‡z 3 associative law of addition;
(c) z 1 z 2 ˆ z 2 z 1
commutative law of multiplication;
(d) z 1 …z 2 z 3 †ˆ…z 1 z 2 †z 3
associative law of multiplication.
6.2. Given
z 1 ˆ 3 ‡ 4i
3 4i ; z 2 ˆ 1 ‡ 2i 2
1 3i
®nd their polar forms, complex conjugates, moduli, product, the quotient
z 1 =z 2 :
6.3. The absolute value or modulus of a complex number z ˆ x ‡ iy is de®ned as
p
jjˆ z
q
zz* ˆ x 2 ‡ y 2 :
If z 1 ; z 2 ; ...; z m are complex numbers, show that the following hold:
(a) jz 1 z 2 jˆjz 1 jjz 2 j or jz 1 z 2 z m jˆjz 1 jjz 2 jjz m j:
(b) jz 1 =z 2 jˆjz 1 j=jz 2 j if z 2 6ˆ 0:
(c) jz 1 ‡ z 2 jjz 1 j‡jz 2 j:
(d) jz 1 ‡ z 2 jjz 1 jjz 2
j or jz 1 z 2 jjz 1 jjz 2 j.
6.4 Find all roots of (a)
5p
32, and (b)
3p
1 ‡ i, and locate them in the complex
plane.
6.5 Show, using De Moivre's theorem, that:
(a) cos 5 ˆ 16 cos 5 20 cos 3 ‡ 5cos;
(b) sin 5 ˆ 5 cos 4 sin 10 cos 2 sin 3 ‡ sin 5 .
6.6 Given z ˆ re i , interpret ze i , where is real geometrically.
6.7 Solve the quadratic equation az 2 ‡ bz ‡ c ˆ 0; a 6ˆ 0.
6.8 A point P moves in a counterclockwise direction around a circle of radius 1
with center at the origin in the z plane. If the mapping function is w ˆ z 2 ,
show that when P makes one complete revolution the image P 0 of P in the w
plane makes three complete revolutions in a counterclockwise direction on a
circle of radius 1 with center at the origin.
6.9 Show that f …z† ˆln z has a branch point at z ˆ 0.
6.10 Let w ˆ f …z† ˆ…z 2 ‡ 1† 1=2 , show that:
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