1) what are Cauchy’s Integral Theorem and Cauchy's integral
formula
2) Explain about the consequences and applications of these
theorems.
3) Explain about different types of singularities and the
difference between Taylor series and Laurent series.
Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite
abelian group and p be a prime such that p divides the order of G
then G has an element of order p.
Problem 2.1. Prove this theorem.
The singularity at the center of a black hole is predicted to be
a region of zero volume and infinite density that contains all of
the black hole's mass. It is a point at which all currently known
physical laws break down. Yet in a black hole, this "terrible
point" is hidden from view behind an event horizon that prevents
any knowledge about the singularity reaching the outside universe.
Astronomers continue to spend considerable effort trying to
understand the nature...
Draw the region and evaluate the following integral(a) ∫10∫21(y+2x)dydx∫01∫12(y+2x)dydx (b) ∫10∫xx2xydydx∫01∫x2xxydydx (c) ∫π0∫πysinxxdxdy∫0π∫yπsinxxdxdy
Use a double integral to find the area of the region. The region
inside the cardioid r = 1 + cos(θ) and outside the circle r = 3
cos(θ). Can someone explain to me where to get the limits of
integration for θ? I get how to get the pi/3 and -pi/3 but most
examples of this problem show further that you have to do more for
the limits of integration but I do not get where they come
from?
a) State Mellin’s inverse Laplace transform formula.
b) State Cauchy’s residue theorem.
iii. Use (a) and (b) to prove that the inverse Laplace transform of
F(s)=1/(s+a) is equal to f(t)= e^(-at),t>0