Aktosun, Spring 2023, Math 5322, Supplementary Problems 4

  1. Evaluate the following integrals:
    1.     ∫-∞∞dx/(x4+1)    solution: π/21/2.
    2.     ∫0∞dx/(x6+1)    solution: π/3.
    3.     ∫0∞[sinx/x]dx    solution: π/2.
    4.     ∫02πdθ/(2+cosθ)    solution: 2π/31/2.
    5.     ∫02πdθ/(5+3sinθ)    solution: π/2.
    6.     ∫-∞∞cos(4x) dx/(x2+1)2    solution: 5πe-4/2.
    7.     ∫-∞∞xsinx dx/(x2+2x+2)    solution: πe-1(cos1+sin1).

  2. Evaluate the following integrals along the indicated path:
    1.     ∫C 3i dz/[z2+4z+1] where C is the unit circle traversed once in the counterclockwise direction    solution: -31/2π.
    2.     ∫C 3i dz/[z2+4z+1] where C is the ellipse x2+4y2=1 traversed once in the counterclockwise direction    solution: -31/2π.
    3.     ∫C 3i dz/[z2+z-2] where C is the unit circle traversed once in the counterclockwise direction    solution: -π (as the Cauchy principal value).
    4.     ∫C dz/z where C is the quarter of the unit circle from z=1 to z=i    solution: πi/2.
    5.     ∫C [(x2-y2)dx-2xydy] where C is the line segment from z=1-i to z=3i    solution: 2/3.
    6.     ∫C 2z dz/(z2+1) where C is the part of the circle |z|=2 running from z=-2 to z=2i in the clockwise direction    solution: -ln(5/3)-πi.
    7.     ∫C 2z dz/(z2+1) where C is the part of the circle |z|=2 running from z=-2 to z=2i in the counterclockwise direction    solution: 3πi-ln(5/3).
    8.     ∫C 2z dz/(z2+1) where C is the circle |z|=2 running in the counterclockwise direction    solution: 4πi.
    9.     ∫C z3 dz/(z4-1) where C is circle |z|=2 running in the counterclockwise direction    solution: 2πi.

  3. Find the number of zeros of the following functions within the curve indicated:
    1.     z7-4z3+z-1 inside the circle |z|=1    solution: 3.
    2.     2z5-6z2+z+1 inside the annular region 1≤|z|<2.    solution: 3.

  4. Describe the following:
    1.     The argument principle
    2.     Laurent series    solution: a power series containing negative powers as well,   ∑n=-∞∞an(z-c)n
    3.     The Cauchy-Riemann equations    solution: ux=vy and uy=-vx
    4.     The fundamental theorem of algebra    solution: Every polynomial of degree n (with n≥1) has at least one zero. Equivalently, every polynomial of degree n has exactly n zeros (some may be repeated).
    5.     Liouville's theorem    solution: If a function is analytic and bounded in the whole complex plane, then that function must be a constant.
    6.     The Cauchy integral formula, Cauchy's integral formula
    7.     Morera's theorem    solution: If a function is continuous in a region and its integral along every closed path in that region is zero, then that function must be analytic in that region.
    8.     Green's theorem
    9.     Euler's formula    solution: eiθ=cosθ+isinθ.
    10.     The residue theorem
    11.     De Moivre's formula, De Moivre's theorem    solution: (cosθ+isinθ)n=cos(nθ)+isin(nθ) for n=1,2,3,....
    12.     The Cauchy integral theorem, Cauchy's integral theorem, the Cauchy-Goursat theorem
    13.     The maximum modulus principle
    14.     Singularity of a function
    15.     Zero of a function    solution: a point in the domain of a function at which the value of the function is zero
    16.     The Cauchy principal value
    17.     Improper integral
    18.     Jordan's lemma
    19.     Rouche′'s theorem
    20.     Conformal mapping
    21.     Schwarz' lemma

  5. Determine all the limit points of the following sequences:
    1.     (-i)n+in/(n+1)    solution: 0,   2i,   1+i, and   -1+i.
    2.     [(1+i)/21/2]n    solution: ±1,   ±i,   (1±i)/21/2,   (-1±i)/21/2.

  6. Determine where the following series converge and where they diverge on the complex plane:
    1.     ∑n=1∞ [1+(-1)n]/zn   solution: diverges for |z|≤1 and converges for |z|>1.
    2.     ∑n=1∞ [i+(-1)n]/z2n   solution: diverges for |z|≤1 and converges for |z|>1.

  7. Prove the following:
    1.     If the function f is represented by the power series ∑n=0∞ an(z-c)n in the region |z-c|<R for some positive R, then f′(z) exists when |z-c|<R and that f′(z)=∑n=1∞ nan(z-c)n-1.


Tuncay Aktosun aktosun@uta.edu
Last modified: November 28, 2022