I have question about Cauchy Integral Theorem.
a) Evaluate integral of c (1/(1+z^2))dz where c is the circle of radius 1 about i, traced counterclockwise.
b)Find all possible values of integral of c (1/(1+z^2))dz where B is some path joining 0 to 1 which is in the domain of f(z) = 1/(1+z^2)
a) I used the Cauchy Integral Formula,
since 1/(1+z^2) = 1/((z+i)(z-i)), so let f(z) = 1/(z+i) and f(z0) = i
therefore (2*pi*i)f(i) = pi
b) I did in two ways.
i)when the B is simply connected, f(z) = g'(z), where g'(z) = arctan(z)
therefore, integral c f(z)dz = g(end of B) - g(beginning of B), which is
arctan(1) - arctan(0) = 0.785398162
ii)let c(t) = Re^(it) and we put this on i
ntegral, we get
integral(from 0 to 2pi) (iRe^(it))/(1+((Re^it)^2)dt
if we do this integral artan(e^it) when t = 0 or 2pi, so
artan(e^2ipi)-arctan(1) = 0.214601837
But it says all possible values...is there anymore answer?
And I am really sorry about all the mass. This is first time using this, so I have no idea where I can find and use the math form and signs.
I will try more and I will write properly next time.
Please help me on this first...it has to be done by tomorrow...