Complex integration

Calculate the integral using contour integration. Complete explanation is required. integral(o->oo)logxdx/(1+x^2)^2

Increasing/decreasing holomorphic functions

1. Let f(z) be a holomorphic function in the disc |z| < R1 and set M(r) = sup|f(z)|(|z|=r), A(r) = supR(f(z)) (|z|

Analytic function

3. Let D = {z : |z| < 1}. Suppose that f : D -> D is analytic, f(1/3) = 0 and f'(1/3) = 0. Show that |f(0)| <= 1/9.

Analytic function

Prove that if f(z) : H -> H is an analytic function from the upper-half plane to itself, then:|f(z) − f(z_0)|/|f(z) − (f(z_0))bar|<=|z − z_0|/|z − (z_0)bar| where z,z_0 are in H and |f'(z)|/Im(f(z))<=1/Im(z) where z is in H. When does equality holds?

Holomorphic map

Suppose z= phi(&) and w=psi(&) are one-to-one analytic maps from the unit disc D(0,1) onto the regions G_1 and G_2. Set phi(0)=z_0 and psi(0)=w_0. Let 0G-2 be holomorphic map with f(z_0)=w_0. Show that f(omega_1(r)) is contained in omega_2(r)

Entire function

Show that if an entire function f maps the real axis into itself and the imaginary axis into itself, then f is an odd function, i.e., f(−z) = −f(z) for any z. Give two proofs, which are really different.

Poles

(a) Let f be analytic in a bounded region D and its boundary C, such that |f(z)| = 1 on C. Show that f has at least one zero inside D, unless f is a constant. (b) Let f(z) be an analytic function in a region D except for one simple pole and assume |f(z)| = 1 on the boundary of D. Prove that every value a with |a| > 1 is take ...continues

Roots of equation

(a) How many roots of the equation z^4 − 6z + 3 = 0 have their modulus between 1 and 2? (b) Find the number of the roots of the equation z^6 − 5z^4 + 8z − 1 = 0 in the annulus {z : 1 < |z| < 2}

Solutions to equation

Let lambda be real and lambda > 1, Show that the equation ze^lambda−z = 1 has exactly one solution in the disc |z| = 1, which is real and positive.

Series

Fix R>0. Show that, if n is large enough, then P_n(z)=1+z+z^2/2!+z^3/3!+...+z^n/n! has no zeros in {z:|z|<=R}