Math 6720
HW1
Appl. Complex Var., Asymptc Mthds
Alexander Balk
NAME: .....................................................................
due Wednesday, 1/20/2016
1. Using the Euler formula, derive the trigonometric identities
(a)
cos(3α) = 4 cos3 α − 3 cos α
{Suggestion:
sin(3α) = −4 sin3 α + 3 sin α
and
Take the real and imaginary part of the identity e3iα = eiα
3
.}
(b)
cos α + cos β = 2 cos
{Suggestion:
α+β
α−β
cos
2
2
and
Take the real and imaginary part of the identity eiα + eiβ = ei
sin α + sin β = 2 sin
α+β
2
α+β
α−β
cos
2
2
α−β
α−β ei 2 + e−i 2
; here the expression in paranthesis is real.}
2. Show that
sin x + sin 2x + . . . + sin N x =
sin N2 x · sin N 2+1 x
sin x2
3. Find all values of 1i .
4. Solve the equation cos z = 3.
5. Find all values of arctan(3i).
{Suggestion:
w = arctan 3i
def
⇔
tan w = 3i;
⇔
sin w
cos w
= 3i;
solve the later equation.}
6. A function w(z) of a complex variable z = x + iy takes only real values.
Can it be dierentiable (in complex sense) at some points?
Can it be analytic?
7. (a) Show: If a function w(z) = u(x, y) + iv(x, y) is analytic in a domain D ∈ C
then u(x, y) and v(x, y) are harmonic functions in that domain.
(z = x + iy),
(b) Show: If a function u(x, y) is harmonic in a simply connected domain D ∈ R2 ,
then there is a function v(x, y), [(x, y) ∈ D], such that u(x, y) + iv(x, y) = w(z) is analytic in D.
(c) Would the last statement hold without the requirement of D being simply connected?