Aktosun, Spring 2023,
Math 5322, Supplementary Problems 1
 Simplify the following:
 (3i^{34}i^{23})/(2i1).
solution:
1+i.
 (112i)^{1/3}.
solution:
125^{1/6}exp(πi/3+ia/3),
125^{1/6}exp(πi/3+ia/3),
and 125^{1/6}exp(ia/3),
where a=tan^{1}(2/11).
 Solve the following equations:
 z^{2}(2i+3)z+5+i=0.
solution:
1i and 2+3i.
 6z^{4}25z^{3}+32z^{2}+3z=10.
solution:
1/2, 2/3, 2±i.
 z^{2}(1z^{2})=16.
solution:
(3±7^{1/2}i)/2 and
(3±7^{1/2}i)/2.
 (1+z)^{5}=(1z)^{5}.
solution:
0, ±i[520^{1/2}]^{1/2},
±i[5+20^{1/2}]^{1/2}.
 Explain how the following basic functions are defined and state whether
each function is single valued, double valued, etc.
 z^{n}, n=0,1,2,....
 1/z^{n}, n=0,1,2,....
 e^{z}.
 trigonometric functions.
 hyperbolic functions.
 log(z).
 inverse trigonometric functions.
 inverse hyperbolic functions.
 z^{w}.
 Prove the following or disprove with a counterexample:
 sin(z)≤1.
 cos(z)≤1.
 sin(z)≤z.
 sin^{2}z+cos^{2}z=1.
 sec^{2}ztan^{2}z=1.
 sin(z_{1}+z_{2})=(sinz_{1})(cosz_{2})+(cosz_{1})(sinz_{2})
 cos(z_{1}+z_{2})=(cosz_{1})(cosz_{2})+(sinz_{1})(sinz_{2})
 z_{1}z_{2}≤z_{1}+z_{2}≤z_{1}+z_{2}.
 z_{1}z_{2}≤z_{1}z_{2}≤z_{1}+z_{2}.
 Re(z)≤z.
 Im(z)≤z.
 z^{n}=z^{n}, n=0,1,2,....
 1/z^{n}=1/z^{n}, n=1,2,....
 e^{z}=e^{Re(z)}.
 Prove the following:
 z_{1}+z_{2}^{2}+z_{1}z_{2}^{2}=2(z_{1}^{2}+z_{2}^{2}).
hint:
use z^{2}=zz^{*}
for each of the complex numbers z_{1}+z_{2},
z_{1}z_{2}, z_{1}, and z_{2}.
 cos(2π/n)+cos(4π/n)+cos(6π/n)+...+cos(2(n1)π/n)=1.
hint:
let w=exp(i2π/n),
use 1+w+w^{2}+...+w^{n1}=(1w^{n})/(1w) with w^{n}=1 and take the real parts.
 sin(2π/n)+sin(4π/n)+sin(6π/n)+...+sin(2(n1)π/n)=0.
hint:
let w=exp(i2π/n),
use 1+w+w^{2}+...+w^{n1}=(1w^{n})/(1w) with w^{n}=1 and take the imaginary parts.
 sin(3x)=3sinx4sin^{3}x.
hint:
use De Moivre's theorem with n=3 and use
the imaginary parts in the resulting equation.
 cos(4x)=8sin^{4}x8sin^{2}x+1.
hint:
use De Moivre's theorem with n=4 and use
the real parts in the resulting equation.
 Describe the following families of curves in the complex plane:
 Re(1/z)=C.
solution:
family of circles centered at 1/(2C)+0i of
radius 1/2C but with the origin excluded.
 Im(1/z)=C.
solution:
family of circles centered at 0i/(2C) of
radius 1/2C but with the origin excluded.
 Re(z^{2})=C.
solution:
family of hyperbolas x^{2}y^{2}=C
on the xy plane
centered at the origin; when C=0 we have the lines
y=±x.
 Im(z^{2})=C.
solution:
family of hyperbolas xy=C/2
on the xy plane
centered at the origin; when C=0 we have the
x and y axes.
 Evaluate the following:
 1+2w+3w^{2}+...+nw^{n1},
where w is an nth root of unity.
solution:
n(n+1)/2 if w=1, n/(w1) if w≠1.
 1+4w+9w^{2}+...+n^{2}w^{n1}
,
where w is an nth root of unity.
solution:
n(2n+1)(n+1)/6 if w=1, [n^{2}w^{2}2n(n+1)w+n(n+2)]/(w1)^{3} if
w≠1.
 For a function f(z), relate the following to each other:
 existence of the derivative of f at z_{0}
 differentiability of f at z_{0}
 differentiability of Re(f) and Im(f)
and the validity of the CauchyRiemann equations at z_{0}
 differentiability of f in z, z* at z_{0} and ∂f/∂z* being zero at z_{0}
 analyticity of f at z_{0}
solution:
1, 2, 3, and 4 are equivalent; 5 holds if and only if the conditions
in any one of 1, 2, 3, and 4 hold also in a neighborhood of z_{0};
for example f(z)=z^{2} is not analytic
at z=0 but it is differentiable at z=0.
 For a function f(z) defined on a domain
W, relate the following to each other:
 existence of the derivative of f on W
 differentiability of f on W
 differentiability of f in z, z* on W and ∂f/∂z* being zero on W
 differentiability of Re(f) and Im(f)
and the validity of the CauchyRiemann equations on W
 analyticity of f on W
solution:
1, 2, 3, 4, and 5 are equivalent.
 For a function f(z) differentiable in z, z* on a domain
W, express ∂f/∂z
and ∂f/∂z* in terms of
the x and y derivatives of Re(f) and Im(f).
solution:
∂f/∂z=(u_{x}+v_{y})/2+i(v_{x}u_{y})/2 and ∂f/∂z*=(u_{x}v_{y})/2+i(v_{x}+u_{y})/2, where u:=Re(f) and v:=Im(f).
 For a function f(z) differentiable on a domain
W, express ∂f/∂z and
∂f/∂z* in terms of
the x and y derivatives of Re(f) and Im(f).
solution:
∂f/∂z=u_{x}+iv_{x}=u_{x}iu_{y}=v_{y}iu_{y}=v_{y}+iv_{x} and ∂f/∂z*=0, where u:=Re(f) and v:=Im(f).
 For a function f(z) differentiable on a domain
W, express df/dz in terms of
the x and y derivatives of Re(f) and Im(f).
solution:
df/dz=u_{x}+iv_{x}=u_{x}iu_{y}=v_{y}iu_{y}=v_{y}+iv_{x}, where u:=Re(f) and v:=Im(f).
 For the following function, study the existence of the derivative
at z=0,
differentiability at z=0, analyticity at z=0,
and validity of the CauchyRiemann equations at z=0:
 f(z)=Re(z).
solution:
f′(0) does not exists;
f is not differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)≠v_{y}(0,0) but
u_{y}(0,0)=v_{x}(0,0)
because we have u_{x}(0,0)=1, u_{y}(0,0)=0,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=Im(z).
solution:
f′(0) does not exists;
f is not differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)=v_{y}(0,0) but
u_{y}(0,0)≠v_{x}(0,0)
because we have u_{x}(0,0)=0, u_{y}(0,0)=1,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=z.
solution:
f′(0) does not exists;
f is not differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)≠v_{y}(0,0) and
u_{y}(0,0)≠v_{x}(0,0)
because u_{x}(0,0) does not exist,
u_{y}(0,0) does not exist,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=z·Re(z).
solution:
f′(0)=0;
f is differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)=v_{y}(0,0) and
u_{y}(0,0)=v_{x}(0,0)
because u_{x}(0,0)=0,
u_{y}(0,0)=0,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=Re(z)·Im(z).
solution:
f′(0)=0;
f is differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)=v_{y}(0,0) and
u_{y}(0,0)=v_{x}(0,0)
because u_{x}(0,0)=0,
u_{y}(0,0)=0,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=z^{2}.
solution:
f′(0)=0;
f is differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)=v_{y}(0,0) and
u_{y}(0,0)=v_{x}(0,0)
because u_{x}(0,0)=0,
u_{y}(0,0)=0,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=z^{4}.
solution:
f′(0)=0;
f is differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)=v_{y}(0,0) and
u_{y}(0,0)=v_{x}(0,0)
because u_{x}(0,0)=0,
u_{y}(0,0)=0,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 f(z)=z^{2}z^{*}.
solution:
f′(0)=0;
f is differentiable at z=0;
f is not analytic at z=0;
u_{x}(0,0)=v_{y}(0,0) and
u_{y}(0,0)=v_{x}(0,0)
because u_{x}(0,0)=0,
u_{y}(0,0)=0,
v_{x}(0,0)=0,
and v_{y}(0,0)=0.
 any polynomial in z.
solution:
the derivative exists everywhere and in particular at
z=0;
differentiable everywhere and in particular at
z=0;
analytic everywhere and in particular at
z=0;
u_{x}=v_{y}
and u_{y}=v_{x} everywhere
and in particular at
z=0.
 sin(z).
solution:
the derivative exists everywhere and in particular at
z=0;
differentiable everywhere and in particular at
z=0;
analytic everywhere and in particular at
z=0;
u_{x}=v_{y}
and u_{y}=v_{x} everywhere
and in particular at
z=0.
 e^{z}.
solution:
the derivative exists everywhere and in particular at
z=0;
differentiable everywhere and in particular at
z=0;
analytic everywhere and in particular at
z=0;
u_{x}=v_{y}
and u_{y}=v_{x} everywhere
and in particular at
z=0.
 1/(z+7+i).
solution:
the derivative exists everywhere except at z=7i,
and in particular it exists at
z=0;
differentiable everywhere except at z=7i,
and in particular differentiable at
z=0;
analytic everywhere except at z=7i,
and in particular analytic at
z=0;
u_{x}=v_{y}
and u_{y}=v_{x} everywhere
except at z=7i,
and in particular these equations hold at z=0.
 For the following function, find the partial derivative with respect to
z, the partial derivative with respect to z^{*}, and the derivative at z=0:
 f(z)=Re(z).
solution:
1/2, 1/2, does not exist.
 f(z)=Im(z).
solution:
i/2, i/2, does not exist.
 f(z)=z.
solution:
does not exist, does not exist, does not exist.
 f(z)=z·Re(z).
solution:
0, 0, 0.
 f(z)=Re(z)·Im(z).
solution:
0, 0, 0.
 f(z)=z^{2}.
solution:
0, 0, 0.
 f(z)=z^{4}.
solution:
0, 0, 0.
 f(z)=z^{2}z^{*}.
solution:
0, 0, 0.
 any polynomial in z.
solution:
the coefficient of z in the polynomial, 0,
the coefficient of z in the polynomial.
 sin(z).
solution:
1, 0, 1.
 e^{z}.
solution:
1, 0, 1.
 1/(z+7+i).
solution:
1/(7+i)^{2}, 0, 1/(7+i)^{2}.
 If z=1i, evaluate the following:
 Arg(z).
solution:
3π/4.
 Arg(z^{2}).
solution:
π/2.
 Arg(1/z).
solution:
3π/4.
 Arg(z^{*}).
solution:
3π/4.
 Arg(1/z^{*}).
solution:
3π/4.
 Arg(1/z^{5}).
solution:
π/4.
 If z=1+i, evaluate the following:
 arg(z).
solution:
3π/4+2πn, where
n is any integer.
 arg(z^{2}).
solution:
π/2+2πn.
 arg(1/z).
solution:
3π/4+2πn.
 arg(z^{*}).
solution:
3π/4+2πn.
 arg(1/z^{*}).
solution:
3π/4+2πn.
 arg(1/z^{5}).
solution:
π/4+2πn.
 If f(z)=x^{2}y^{2}2y+2ix2ixy, find the following:
 ∂f/∂z.
solution:
2i.
 ∂f/∂z^{*}.
solution:
2z^{*}.
 df/dz.
solution:
2i at z=0, does not exist elsewhere.
 all the points at which f is analytic.
solution:
nowhere analytic.
 all the points at which f is differentiable.
solution:
only at z=0.
 all the points at which f is continuous.
solution:
everywhere continuous.
 If g(z)=x^{3}3xy^{2}+x+1+3ix^{2}y+iyiy^{3}, find the following:
 ∂g/∂z.
solution:
3z^{2}+1.
 ∂g/∂z^{*}.
solution:
0.
 dg/dz.
solution:
3z^{2}+1.
 all the points at which g is analytic.
solution:
everywhere analytic.
 all the points at which g is differentiable.
solution:
everywhere differentiable.
 all the points at which g is continuous.
solution:
everywhere continuous.
 If f(z)=(x^{2}y+xy^{2})/(x^{3}+y^{3}) for z≠0 and
f(z)=0 for z=0, find the following:
 u_{x}(0,0), where
u(x,y)=Re[f(z)].
solution:
u_{x}(0,0)=0.
 v_{x}(0,0),
where v(x,y)=Im[f(z)].
solution:
v_{x}(0,0)=0.

is u(x,y) continuous at (0,0)?
solution:
not continuous.

is u(x,y) differentiable at (0,0)?
solution:
not differentiable.

is v(x,y) continuous at (0,0)?
solution:
continuous.

is v(x,y) differentiable at (0,0)?
solution:
differentiable.
 is f continuous at z=0?
solution:
not continuous.
 is f differentiable at z=0?
solution:
not differentiable.
 do the CauchyRiemann equations hold at (0,0)?
solution:
the CauchyRiemann equations hold at (0,0).
 is f analytic at z=0?
solution:
not analytic.
 If f(z)=(x^{2}yxy^{2})/(x^{2}+y^{2}) for z≠0 and
f(z)=0 for z=0, find the following:
 u_{x}(0,0), where
u(x,y)=Re[f(z)].
solution:
u_{x}(0,0)=0.
 v_{x}(0,0),
where v(x,y)=Im[f(z)].
solution:
u_{x}(0,0)=0.

is u(x,y) continuous at (0,0)?
solution:
continuous.

is u(x,y) differentiable at (0,0)?
solution:
not differentiable.

is v(x,y) continuous at (0,0)?
solution:
continuous.

is v(x,y) differentiable at (0,0)?
solution:
differentiable.
 is f continuous at z=0?
solution:
continuous.
 is f differentiable at z=0?
solution:
not differentiable.
 do the CauchyRiemann equations hold at (0,0)?
solution:
the CauchyRiemann equations hold at (0,0).
 is f analytic at z=0?
solution:
not analytic.
 Prove the following:
 z_{1}+z_{2}≤z_{1}+z_{2}
with the equality holding if and only of the real part of z_{1}z_{2}^{*}
is nonnegative.
 the CauchyRiemann equations u_{x}=v_{y} and
u_{y}=v_{x} in the rectangular form can be written as
u_{r}=(1/r)v_{θ} and
v_{r}=(1/r)u_{θ} in the polar form.
Tuncay Aktosun
aktosun@uta.edu
Last modified: February 16, 2022