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Quiz 9 (Notes, books, and calculators are not authorized) Show all your work in the blank space
you are given on the exam sheet. Answers with no justification will not be graded.
Question 1: Let z = x + iy, were x, y ∈ R. Is the function f (z) = cos(y) + i sin(x) holomorphic
in C? (Justify rigorously your answer.)
Let P (x, y) = cos(y) and Q(x, y) = sin(x) be the real and imaginary parts of f . The functions P
and Q are twice continuously differentiable functions with respect to x and y over R2 . Let us verify
whether the Cauchy Riemann relations hold:
∂x P (x, y) = 0,
∂y P (x, y) = − sin(y),
∂y Q(x, y) = 0.
∂y Q(x, y) = cos(x).
The function f is not holomorphic since ∂y P (x, y) 6= −∂x Q(x, y).
Question 2: Using the notation z = reiθ , were r ≥ 0, θ ∈ [−π, π), we define the function
log(z) as follows log(z) := log(r) + iθ. Prove that log(z) is not continuous across the axis
{y = 0, x < 0}.
Let −r0 be an arbitrary number on the negative real axis {y = 0, x < 0}. Let us set z+ (π − ) =
r0 ei(π−) , z− () = r0 ei(−π+) with > 0. It is clear that
lim z+ () = −r0 = lim z− (),
→0
→0
i.e., both z+ () and z− () converge to −r0 as goes to zero. If the complex log function defined
above was continuous, the two quantities log(z+ ) and log(z− ) should converge to the same value
as goes to zero. But using the above definition of log(z) we have
lim log(z+ ) = lim log(r0 ) + i(π − ) = log(r0 ) + iπ
→0
→0
and
lim log(z− ) = lim log(r0 ) + i(−π + ) = log(r0 ) − iπ.
→0
→0
In conclusion lim→0 log(z+ ) 6= lim→0 log(z− ), thereby proving that log(z) as defined above is not
continuous across the axis {y = 0, x < 0}.
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Quiz 9, November 25, 2014
Question 3: Let z = x + iy, were x, y ∈ R. Let ez be defined by ez := ex (cos(y) + i sin(y)).
(a) Prove that eiz = e−iz by using the definition of ez .
By definition
eiz = eix−y = e−y (cos(x) + i sin(x)) = e−y (cos(x) − i sin(x)) = e−y−ix = e−i(x−iy) .
As a result eiz = e−iz
(b) Define cos(z) :=
eiz +e−iz
.
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Compute cos(iy) for y ∈ R.
Using the definition of cos(z), we infer that
cos(iy) =
1 −y
(e + ey ) = cosh(y).
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Question 4: (a) Find the derivative of f (z) = (3 − 2i)z 3 + 2iz − 5 + i and specify where f is
holomorphic
f 0 (z) = 3(3 − 2i)z 2 + 2i.
f is holomorphic in C.
(b) Find the derivative of F (z) =
F 0 (z) =
F is holomorphic in C \ { 21 i}.
3z+i
2z−i
and specify where F is holomorphic.
3(2z − i) − 2(3z + i)
5i
=−
.
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(2z − i)
(2z − i)2