Aktosun, Spring 2023, Math 5322, Supplementary Problems 3

  1. Evaluate the following integrals along the unit circle C centered at the origin traversed in the counterclockwise direction:
    1.     ∫C z* dz    solution: 2πi.
    2.     ∫C (1/z*) dz    solution: 0.
    3.     ∫C (zz*) dz    solution: 0.
    4.     ∫C z2 dz    solution: 0.
    5.     ∫C (1/z2) dz    solution: 0.
    6.     ∫C [z/(z-1/2)2] dz    solution: 2πi.
    7.     ∫C [z/(z-2)2] dz    solution: 0.
    8.     ∫C [1/(2z2+1)] dz    solution: 0.
    9.     ∫C [1/(16z4-1)] dz    solution: 0.
    10.     ∫C [zez/(z-i/3)4] dz    solution: π(i-1/9)ei/3.

  2. Obtain the Maclaurin series representations for the following functions:
    1.     ez    solution: ∑n=0∞[zn/n!]   with   |z|<+∞.
    2.     sinz    solution: ∑n=0∞ [(-1)nz2n+1/(2n+1)!]   with   |z|<+∞.
    3.     cosz    solution: ∑n=0∞ [(-1)nz2n/(2n)!]   with   |z|<+∞.
    4.     tanz    solution: z+z3/3+2z5/15+O(z7)   with   |z|<π/2.
    5.     coshz    solution: ∑n=0∞ [z2n/(2n)!]   with   |z|<∞.
    6.     1/(1+z)    solution: ∑n=0∞ [(-1)nzn]   with   |z|<1.
    7.     1/(1+z)2    solution: ∑n=0∞ [(-1)n(n+1)zn]   with   |z|<1.
    8.     1/(1+z)3    solution: (1/2)∑n=0∞ [(-1)n(n+2)(n+1)zn]   with   |z|<1.
    9.     (1-z)/(1+z)    solution: 1+2∑n=0∞ [(-1)n+1zn+1]   with   |z|<1.
    10.     tanhz    solution: z-z3/3+O(z4)   with   |z|<π/2.

  3. Obtain the Taylor series for the following:
    1.     1/(1+z) about the point z=i    solution: ∑n=0∞[(-1)n(z-i)n/(1+i)n+1]   with   |z-i|<√(2).
    2.     1/z about the point z=3    solution: ∑n=0∞ [(-1)n(z-3)n/3n+1]   with   |z-3|<3.

  4. Obtain the Laurent series in a deleted neighborhood of z=0 for the following:
    1.     z4e1/z    solution: z4+z3+z2/2+z/6+1/24+∑n=0∞ [1/{(n+5)! zn+1}]   with   |z|>0.
    2.     1/[z2(1-z)]    solution: 1/z2+1/z+∑n=0∞[zn]   with   0<|z|<1.
    3.     e1/z/z4    solution: ∑n=0∞[1/{n!zn+4}]   with   |z|>0.

  5. Obtain the Laurent series in a deleted neighborhood of z=∞ for the following:
    1.     1/(z+1)    solution: ∑n=0∞ [(-1)n/zn+1]   with   |z|>1.
    2.     1/[z2(1-z)]    solution: -∑n=0∞ [1/zn+3]   with   |z|>1.
    3.     z/(1-z)    solution: ∑n=0∞ [-1/zn]   with   |z|>1.

  6. Find the residue at z=0 for the following functions:
    1.     1/[z(z-2)3]    solution: -1/8.
    2.     (sinz)/[z2(z2+1)]    solution: 1.
    3.     ez/z2    solution: 1.

  7. Evaluate the following:
    1.     ∫C [1/{z(z-2)4}] dz where C is the circle |z-2|=1 traversed counterclockwise    solution: -πi/8.
    2.     ∫C [(3z-2)/{z(z-4)}] dz where C is the circle |z|=2 traversed counterclockwise    solution: πi.
    3.     ∫C [z4/(1-z3)] dz where C is the circle |z|=2 in the positive direction    solution: 0.
    4.     ∫C [1/z] dz where C is the circle |z|=2 in the positive direction    solution: 2πi.
    5.     ∫C [sin(z)/z] dz where C is the circle |z|=2 in the positive direction    solution: 0.
    6.     ∫C tan(z) dz where C is the circle |z|=2 in the positive direction    solution: -4πi.
    7.     ∫C csch(2z) dz where C is the circle |z|=2 in the positive direction    solution: -πi.
    8.     ∫C [(2z+3)/{z(z-1)(z-2)}] dz where C is the circle |z|=3 in the positive direction    solution: 0.
    9.     ∫C z3e1/z dz where C is the circle |z|=3 in the positive direction    solution: πi/12.

  8. Find the poles and their residues for the following functions:
    1.     (3z2+1)/(z2+z)    solution: simple pole at z=0 with residue 1, simple pole at z=-1 with residue -4.
    2.     sechz    solution: simple poles at z=(n+1/2)πi with residue (-1)n+1i, where  n=0,±1,±2,....
    3.     z/(z4-16)    solution: simple pole at z=2 with residue 1/16, simple pole at z=-2 with residue 1/16, simple pole at z=-2i with residue -1/16, simple pole at z=2i with residue -1/16.
    4.     (sinz)/(z2+1)    solution: simple pole at z=-i with residue sinh(1)/2, simple pole at z=i with residue sinh(1)/2.


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