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1 Questions
- Consider
Z
Γ
x + y
x
2 + y
2
dx +
−x + y
x
2 + y
2
dy.
(a) Calculate the curve integral, where the curve Γ goes in the half-plane y ≥ 0
from the point (1, 0) to (−1, 0) along the superellipse x
6 + 3y
6 = 1.
(b) List all the statements you have used in (a) and explain how.
Note: If in the formulation of the statement, e.g. a field F, then indicate
how you have chosen F in the example,and explain why all the set of prerequisites are met.
(c) Enter a curve Γ such that the curve integral becomes zero and justify your
choice. - (a) Consider the series
X∞
n=0
2n
n
z
n
. (1)
i. Determine the radius of convergence of the series.
ii. Enter an amount of M ⊂ C such that the series converges uniformly in
M.
iii. The new concepts of uniform convergent function sequence and uniform
convergent function series. Explain the difference between point-by-point and
uniform convergence pursuits.
(b) Consider the series
X∞
n=1
(−1)n
n · 9
n
z
2n
.
i. Determine the radius of convergence of the series R.
ii. Determine the value of the series for |z| < R.
iii. Does the series also converge for all z C with |z| = R?
Motivate your answers!
1
Sample Solution
