Complex Analysis and Singularities : If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

Complex Analysis / Singularities : Now, prove that a function f has an essential singularity at z = a iff neither of the above holds for any real number s.

One can classify isolated singularities by examining the equations: lim (z -> a) |z - a|^s |f(z)| = 0 lim(z -> a) |z - a|^s |f(z)| = infinity Now, prove that a function f has an essential singularity at z = a iff neither of the above holds for any real number s.

Residues, Circles and Poles

Find the residues of the following. f(z) =(z2 + 4)/(z3 + 2z2 +2z) f(z)= (z+1)^2/ (z-1)^2 Define residues and give formulae to calculate residues. Please see the attachment

Complex Analysis / Singularities / Argument Principle : Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G.

Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G. In your solution, please refer to theorems or certain lemmas. Justify your claims and steps. I want to learn not just have the right answer. Thanks.

Complex Analysis / Argument Principle : Analyticity, Solutions and Multiplicities

Suppose f is analytic on B(bar) (0;1) and satisfies |f(z)| < 1 for |z| = 1. Find the number of solutions (counting multiplicities) of the equation f(z) = z^n, where n is an integer larger than or equal to 1. Please justify every step and claim and refer to any theorems you use.

Argument Principle / Complex Analysis : Nonconstant Meromorphic Functions and Mapping to the Complex Plane

Is a nonconstant meromorphic function on a region G an open mapping of G into C? Is it an open mapping of G into C_oo ( C is complex plane, oo infinity, C_oo means the extended complex plane ( C U {oo} ) ).

Argument Principle / Complex Analysis : Let f be analytic in a neighborhood of D = B(bar)(0;1). If |f(z)|=<1 for |z|=1, what can you say?

Let f be analytic in a neighborhood of D = B(bar)(0;1). If |f(z)|=<1 for |z|=1, what can you say? Justify every step and claim and refer to any theorems you use please.

Residues / Integrals - Complex Analysis : integral from 0 to pi/2 of ( d theta/ ( a + sin^2 theta) ) = pi/2[a(a+1)]^1/2 if a > 0.

Verify the following equation: integral from 0 to pi/2 of ( d theta/ ( a + sin^2 theta) ) = pi/2[a(a+1)]^1/2 if a > 0.

Maximum Modulus Theorem : Let f be analytic in the disk B(0;R) and for 0 =< r < R define A(r) = max { Re f(z) : |z| = r}. Show that unless f is a constant, A(r) is a strictly increasing function of r.

Let f be analytic in the disk B(0;R) and for 0 =< r < R define A(r) = max { Re f(z) : |z| = r}. Show that unless f is a constant, A(r) is a strictly increasing function of r. Please justify every step and claim and show how you used all what is given. Also refer to theorems or lemmas used in the proof. The section where I ...continues

Harmonic Function : Give an example (and explain why it works) of an analytic function u on a harmonic function v such that the composite function u o v is defined but NOT harmonic.

Give an example (and explain why it works) of an analytic function u on a harmonic function v such that the composite function u o v is defined but NOT harmonic. Please see the attached file for the fully formatted problem.