Aktosun, Spring 2023,
Math 5322, Supplementary Problems 4
 Evaluate the following integrals:

∫_{∞}^{∞}dx/(x^{4}+1)
solution:
π/2^{1/2}.

∫_{0}^{∞}dx/(x^{6}+1)
solution:
π/3.

∫_{0}^{∞}[sinx/x]dx
solution:
π/2.

∫_{0}^{2π}dθ/(2+cosθ)
solution:
2π/3^{1/2}.

∫_{0}^{2π}dθ/(5+3sinθ)
solution:
π/2.

∫_{∞}^{∞}cos(4x) dx/(x^{2}+1)^{2}
solution:
5πe^{4}/2.

∫_{∞}^{∞}xsinx dx/(x^{2}+2x+2)
solution:
πe^{1}(cos1+sin1).
 Evaluate the following integrals along the indicated
path:

∫_{C}
3i dz/[z^{2}+4z+1]
where C is the unit circle traversed once in the counterclockwise
direction
solution:
3^{1/2}π.

∫_{C}
3i dz/[z^{2}+4z+1]
where C is the ellipse x^{2}+4y^{2}=1
traversed once in the counterclockwise
direction
solution:
3^{1/2}π.

∫_{C}
3i dz/[z^{2}+z2]
where C is the unit circle traversed once in the counterclockwise
direction
solution:
π (as the Cauchy principal value).

∫_{C}
dz/z
where C is the quarter of the unit circle
from z=1 to z=i
solution:
πi/2.

∫_{C}
[(x^{2}y^{2})dx2xydy]
where C is the line segment from z=1i to z=3i
solution:
2/3.

∫_{C}
2z dz/(z^{2}+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.

∫_{C}
2z dz/(z^{2}+1)
where C is the part of the circle
z=2 running from z=2 to z=2i
in the counterclockwise direction
solution:
3πiln(5/3).

∫_{C}
2z dz/(z^{2}+1)
where C is the circle z=2 running
in the counterclockwise direction
solution:
4πi.

∫_{C}
z^{3} dz/(z^{4}1)
where C is circle z=2 running
in the counterclockwise direction
solution:
2πi.
 Find the number of zeros of the following functions
within the curve indicated:

z^{7}4z^{3}+z1 inside
the circle z=1
solution:
3.

2z^{5}6z^{2}+z+1 inside
the annular region 1≤z<2.
solution:
3.
 Describe the following:

The argument principle

Laurent series
solution:
a power series containing negative powers as well,
∑_{n=∞}^{∞}a_{n}(zc)^{n}

The CauchyRiemann equations
solution:
u_{x}=v_{y} and
u_{y}=v_{x}

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).

Liouville's theorem
solution:
If a function is analytic and bounded in the whole complex plane, then that
function must be a constant.

The Cauchy integral formula, Cauchy's integral formula

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.

Green's theorem

Euler's formula
solution:
e^{iθ}=cosθ+isinθ.

The residue theorem

De Moivre's formula, De Moivre's theorem
solution:
(cosθ+isinθ)^{n}=cos(nθ)+isin(nθ) for n=1,2,3,....

The Cauchy integral theorem, Cauchy's integral theorem,
the CauchyGoursat theorem

The maximum modulus principle

Singularity of a function

Zero of a function
solution:
a point in the domain of a function at which the value of the function is zero

The Cauchy principal value

Improper integral

Jordan's lemma

Rouche′'s theorem

Conformal mapping

Schwarz' lemma
 Determine all the limit points of the following sequences:

(i)^{n}+in/(n+1)
solution:
0, 2i, 1+i, and 1+i.

[(1+i)/2^{1/2}]^{n}
solution:
±1,
±i,
(1±i)/2^{1/2}, (1±i)/2^{1/2}.
 Determine where the following series converge and where they diverge
on the complex plane:

∑_{n=1}^{∞
}[1+(1)^{n}]/z^{n}
solution:
diverges for z≤1 and
converges for z>1.

∑_{n=1}^{∞
}[i+(1)^{n}]/z^{2n}
solution:
diverges for z≤1 and
converges for z>1.
 Prove the following:

If the function f is represented by the
power series ∑_{n=0}^{∞
}a_{n}(zc)^{n}
in the region zc<R for
some positive R, then
f′(z) exists
when zc<R and that
f′(z)=∑_{n=1}^{∞
}na_{n}(zc)^{n1}.
Tuncay Aktosun
aktosun@uta.edu
Last modified: November 28, 2022