Aktosun, Spring 2023, Math 5322, Supplementary Problems 1

  1. Simplify the following:
    1.     (3i34-i23)/(2i-1).    solution: 1+i.
    2.     (-11-2i)1/3.    solution: 1251/6exp(πi/3+ia/3), 1251/6exp(-πi/3+ia/3), and -1251/6exp(ia/3), where a=tan-1(2/11).

  2. Solve the following equations:
    1.     z2-(2i+3)z+5+i=0.    solution: 1-i and 2+3i.
    2.     6z4-25z3+32z2+3z=10.    solution: -1/2, 2/3, 2±i.
    3.     z2(1-z2)=16.    solution: (3±71/2i)/2 and (-3±71/2i)/2.
    4.     (1+z)5=(1-z)5.    solution: 0, ±i[5-201/2]1/2, ±i[5+201/2]1/2.

  3. Explain how the following basic functions are defined and state whether each function is single valued, double valued, etc.
    1.     zn,  n=0,1,2,....
    2.     1/zn,  n=0,1,2,....
    3.     ez.
    4.     trigonometric functions.
    5.     hyperbolic functions.
    6.     log(z).
    7.     inverse trigonometric functions.
    8.     inverse hyperbolic functions.
    9.     zw.

  4. Prove the following or disprove with a counterexample:
    1.     |sin(z)|≤1.
    2.     |cos(z)|≤1.
    3.     |sin(z)|≤|z|.
    4.     sin2z+cos2z=1.
    5.     sec2z-tan2z=1.
    6.     sin(z1+z2)=(sinz1)(cosz2)+(cosz1)(sinz2)
    7.     cos(z1+z2)=(cosz1)(cosz2)+(sinz1)(sinz2)
    8.     ||z1|-|z2||≤|z1+z2|≤|z1|+|z2|.
    9.     ||z1|-|z2||≤|z1-z2|≤|z1|+|z2|.
    10.     |Re(z)|≤|z|.
    11.     |Im(z)|≤|z|.
    12.     |zn|=|z|n,  n=0,1,2,....
    13.     |1/zn|=1/|z|n,  n=1,2,....
    14.     |ez|=eRe(z).

  5. Prove the following:
    1.     |z1+z2|2+|z1-z2|2=2(|z1|2+|z2|2).    hint: use |z|2=zz* for each of the complex numbers z1+z2, z1-z2, z1, and z2.
    2.     cos(2π/n)+cos(4π/n)+cos(6π/n)+...+cos(2(n-1)π/n)=-1.    hint: let w=exp(i2π/n), use 1+w+w2+...+wn-1=(1-wn)/(1-w) with wn=1 and take the real parts.
    3.     sin(2π/n)+sin(4π/n)+sin(6π/n)+...+sin(2(n-1)π/n)=0.    hint: let w=exp(i2π/n), use 1+w+w2+...+wn-1=(1-wn)/(1-w) with wn=1 and take the imaginary parts.
    4.     sin(3x)=3sinx-4sin3x.    hint: use De Moivre's theorem with n=3 and use the imaginary parts in the resulting equation.
    5.     cos(4x)=8sin4x-8sin2x+1.    hint: use De Moivre's theorem with n=4 and use the real parts in the resulting equation.

  6. Describe the following families of curves in the complex plane:
    1.     Re(1/z)=C.    solution: family of circles centered at 1/(2C)+0i of radius 1/|2C| but with the origin excluded.
    2.     Im(1/z)=C.    solution: family of circles centered at 0-i/(2C) of radius 1/|2C| but with the origin excluded.
    3.     Re(z2)=C.    solution: family of hyperbolas x2-y2=C on the xy plane centered at the origin; when C=0 we have the lines y=±x.
    4.     Im(z2)=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.

  7. Evaluate the following:
    1.     1+2w+3w2+...+nwn-1,   where w is an nth root of unity.    solution: n(n+1)/2 if w=1, n/(w-1) if w≠1.
    2.     1+4w+9w2+...+n2wn-1 ,   where w is an nth root of unity.    solution: n(2n+1)(n+1)/6 if w=1, [n2w2-2n(n+1)w+n(n+2)]/(w-1)3 if w≠1.

  8. For a function f(z), relate the following to each other:
    1.     existence of the derivative of f at z0
    2.     differentiability of f at z0
    3.     differentiability of Re(f) and Im(f) and the validity of the Cauchy-Riemann equations at z0
    4.     differentiability of f in z, z* at z0 and ∂f/∂z* being zero at z0
    5.     analyticity of f at z0
         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 z0; for example f(z)=|z|2 is not analytic at z=0 but it is differentiable at z=0.

  9. For a function f(z) defined on a domain W, relate the following to each other:
    1.     existence of the derivative of f on W
    2.     differentiability of f on W
    3.     differentiability of f in z, z* on W and ∂f/∂z* being zero on W
    4.     differentiability of Re(f) and Im(f) and the validity of the Cauchy-Riemann equations on W
    5.     analyticity of f on W
         solution: 1, 2, 3, 4, and 5 are equivalent.

  10. 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=(ux+vy)/2+i(vx-uy)/2   and   ∂f/∂z*=(ux-vy)/2+i(vx+uy)/2,   where u:=Re(f) and v:=Im(f).

  11. 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=ux+ivx=ux-iuy=vy-iuy=vy+ivx   and   ∂f/∂z*=0,   where u:=Re(f) and v:=Im(f).

  12. 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=ux+ivx=ux-iuy=vy-iuy=vy+ivx,   where u:=Re(f) and v:=Im(f).

  13. 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 Cauchy-Riemann equations at z=0:
    1.     f(z)=Re(z).    solution: f′(0) does not exists; f is not differentiable at z=0; f is not analytic at z=0; ux(0,0)≠vy(0,0) but uy(0,0)=-vx(0,0) because we have ux(0,0)=1, uy(0,0)=0, vx(0,0)=0, and vy(0,0)=0.
    2.     f(z)=Im(z).    solution: f′(0) does not exists; f is not differentiable at z=0; f is not analytic at z=0; ux(0,0)=vy(0,0) but uy(0,0)≠-vx(0,0) because we have ux(0,0)=0, uy(0,0)=1, vx(0,0)=0, and vy(0,0)=0.
    3.     f(z)=|z|.    solution: f′(0) does not exists; f is not differentiable at z=0; f is not analytic at z=0; ux(0,0)≠vy(0,0) and uy(0,0)≠-vx(0,0) because ux(0,0) does not exist, uy(0,0) does not exist, vx(0,0)=0, and vy(0,0)=0.
    4.     f(z)=z·Re(z).    solution: f′(0)=0; f is differentiable at z=0; f is not analytic at z=0; ux(0,0)=vy(0,0) and uy(0,0)=-vx(0,0) because ux(0,0)=0, uy(0,0)=0, vx(0,0)=0, and vy(0,0)=0.
    5.     f(z)=|Re(z)·Im(z)|.    solution: f′(0)=0; f is differentiable at z=0; f is not analytic at z=0; ux(0,0)=vy(0,0) and uy(0,0)=-vx(0,0) because ux(0,0)=0, uy(0,0)=0, vx(0,0)=0, and vy(0,0)=0.
    6.     f(z)=|z|2.    solution: f′(0)=0; f is differentiable at z=0; f is not analytic at z=0; ux(0,0)=vy(0,0) and uy(0,0)=-vx(0,0) because ux(0,0)=0, uy(0,0)=0, vx(0,0)=0, and vy(0,0)=0.
    7.     f(z)=|z|4.    solution: f′(0)=0; f is differentiable at z=0; f is not analytic at z=0; ux(0,0)=vy(0,0) and uy(0,0)=-vx(0,0) because ux(0,0)=0, uy(0,0)=0, vx(0,0)=0, and vy(0,0)=0.
    8.     f(z)=z2z*.    solution: f′(0)=0; f is differentiable at z=0; f is not analytic at z=0; ux(0,0)=vy(0,0) and uy(0,0)=-vx(0,0) because ux(0,0)=0, uy(0,0)=0, vx(0,0)=0, and vy(0,0)=0.
    9.     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; ux=vy and uy=-vx everywhere and in particular at z=0.
    10.     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; ux=vy and uy=-vx everywhere and in particular at z=0.
    11.     ez.    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; ux=vy and uy=-vx everywhere and in particular at z=0.
    12.     1/(z+7+i).    solution: the derivative exists everywhere except at z=-7-i, and in particular it exists at z=0; differentiable everywhere except at z=-7-i, and in particular differentiable at z=0; analytic everywhere except at z=-7-i, and in particular analytic at z=0; ux=vy and uy=-vx everywhere except at z=-7-i, and in particular these equations hold at z=0.

  14. 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:
    1.     f(z)=Re(z).    solution: 1/2, 1/2, does not exist.
    2.     f(z)=Im(z).    solution: -i/2, i/2, does not exist.
    3.     f(z)=|z|.    solution: does not exist, does not exist, does not exist.
    4.     f(z)=z·Re(z).    solution: 0, 0, 0.
    5.     f(z)=|Re(z)·Im(z)|.    solution: 0, 0, 0.
    6.     f(z)=|z|2.    solution: 0, 0, 0.
    7.     f(z)=|z|4.    solution: 0, 0, 0.
    8.     f(z)=z2z*.    solution: 0, 0, 0.
    9.     any polynomial in z.    solution: the coefficient of z in the polynomial, 0, the coefficient of z in the polynomial.
    10.     sin(z).    solution: 1, 0, 1.
    11.     ez.    solution: 1, 0, 1.
    12.     1/(z+7+i).    solution: -1/(7+i)2, 0, -1/(7+i)2.

  15. If   z=-1-i, evaluate the following:
    1.     Arg(z).    solution: -3π/4.
    2.     Arg(z2).    solution: π/2.
    3.     Arg(1/z).    solution: 3π/4.
    4.     Arg(z*).    solution: 3π/4.
    5.     Arg(1/z*).    solution: -3π/4.
    6.     Arg(1/z5).    solution: -π/4.

  16. If   z=-1+i, evaluate the following:
    1.     arg(z).    solution: 3π/4+2πn, where n is any integer.
    2.     arg(z2).    solution: -π/2+2πn.
    3.     arg(1/z).    solution: -3π/4+2πn.
    4.     arg(z*).    solution: -3π/4+2πn.
    5.     arg(1/z*).    solution: 3π/4+2πn.
    6.     arg(1/z5).    solution: π/4+2πn.

  17. If   f(z)=x2-y2-2y+2ix-2ixy, find the following:
    1.     ∂f/∂z.    solution: 2i.
    2.     ∂f/∂z*.    solution: 2z*.
    3.     df/dz.    solution: 2i at z=0, does not exist elsewhere.
    4.     all the points at which f is analytic.    solution: nowhere analytic.
    5.     all the points at which f is differentiable.    solution: only at z=0.
    6.     all the points at which f is continuous.    solution: everywhere continuous.

  18. If   g(z)=x3-3xy2+x+1+3ix2y+iy-iy3, find the following:
    1.     ∂g/∂z.    solution: 3z2+1.
    2.     ∂g/∂z*.    solution: 0.
    3.     dg/dz.    solution: 3z2+1.
    4.     all the points at which g is analytic.    solution: everywhere analytic.
    5.     all the points at which g is differentiable.    solution: everywhere differentiable.
    6.     all the points at which g is continuous.    solution: everywhere continuous.

  19. If   f(z)=(x2y+xy2)/(x3+y3) for z≠0 and   f(z)=0 for z=0, find the following:
    1.     ux(0,0), where   u(x,y)=Re[f(z)].    solution: ux(0,0)=0.
    2.     vx(0,0),   where v(x,y)=Im[f(z)].    solution: vx(0,0)=0.
    3.     is   u(x,y) continuous at (0,0)?    solution: not continuous.
    4.     is   u(x,y) differentiable at (0,0)?    solution: not differentiable.
    5.     is   v(x,y) continuous at (0,0)?    solution: continuous.
    6.     is   v(x,y) differentiable at (0,0)?    solution: differentiable.
    7.     is   f continuous at z=0?    solution: not continuous.
    8.     is   f differentiable at z=0?    solution: not differentiable.
    9.     do the Cauchy-Riemann equations hold at (0,0)?    solution: the Cauchy-Riemann equations hold at (0,0).
    10.     is   f analytic at z=0?    solution: not analytic.

  20. If   f(z)=(x2y-xy2)/(x2+y2) for z≠0 and   f(z)=0 for z=0, find the following:
    1.     ux(0,0), where   u(x,y)=Re[f(z)].    solution: ux(0,0)=0.
    2.     vx(0,0),   where v(x,y)=Im[f(z)].    solution: ux(0,0)=0.
    3.     is   u(x,y) continuous at (0,0)?    solution: continuous.
    4.     is   u(x,y) differentiable at (0,0)?    solution: not differentiable.
    5.     is   v(x,y) continuous at (0,0)?    solution: continuous.
    6.     is   v(x,y) differentiable at (0,0)?    solution: differentiable.
    7.     is   f continuous at z=0?    solution: continuous.
    8.     is   f differentiable at z=0?    solution: not differentiable.
    9.     do the Cauchy-Riemann equations hold at (0,0)?    solution: the Cauchy-Riemann equations hold at (0,0).
    10.     is   f analytic at z=0?    solution: not analytic.

  21. Prove the following:
    1.     |z1+z2|≤|z1|+|z2| with the equality holding if and only of the real part of z1z2* is nonnegative.
    2.     the Cauchy-Riemann equations ux=vy and uy=-vx in the rectangular form can be written as ur=(1/r)vθ and vr=-(1/r)uθ in the polar form.


Tuncay Aktosun aktosun@uta.edu
Last modified: February 16, 2022