Open mapping theorem for complex variables

Suppose that f is folomorphic in a region G(i.e. an open connected set). How can I prove that in any of the following cases a)R(f) is constant b)I(f) is constant c)|f| is constant d) arg(f) is constant we can conclude that f is constant. Ps. here R(f) and I(f) are the real and imaginary parts of f I think that this might be an application of the open mapping theorem but I don't know how to conclude the above.

Convergence of power series. Unit Circle convergence and convergence of summations of series.

a) Prove that sum(z^n/n) converges at every point of the unit circle except z=1 although this power series has R=1. b) Use partial fractions to determine the following closed expression for c_n c_n=((1+sqrt5/2)^n+1 - (1-sqrt5/2)^n+1)/sqrt5 Ps. Here c_n are Fibonacci numbers defined by c_0=1, c_1=1,.... c_n=c_n-1 + c_n-2, for n=2,3.....

Integration of functions of complex variables

a) compute the integral of xdz (|z|=r) for the positive sense of the circle in two ways first by using parametrization and second by observing that x=(1/2)(z+z conjugate)=(1/2)(z+r^2/z) on the circle. b) Compute the integral of dz/(z^2-1) (|z|=2) for the positive sense of the circle. PS - Here maybe we have to find first a primitive function of the integrant..

Fractional transformation, cross ratio, conformal mapping

1. a) Let z1,z2,z3,z4 lie on a circle. Show that z1,z3,z4 and z2,z3,z4 determine the same orientation iff (z1,z2,z,3,z4)>0 b) Let z1,z2,z3,z4 lie on a circle and be consecutive vertices of a quadrilateral. Prove that |z1-z3|*|z2-z4|=|z1-z2|*|z3-z4|+|z2-z3|*|z1-z4|

Harmonic functions

a)Let a be less than b and set M(z)=(z-ia)/(z-ib). Define the lines L1={z:F(z)=b}, L2={z:F(z)=a} and L3={z:R(z)=0}. The three lines split the complex plane into 6 regions. Determine the image of them in the complex plane. b) Let log be principal branch of the logarithm. Show that log(M(z)) is defined for all z in C with the exception of the linesegment from ia to ib. c) Define h(z)=F(log(M(z))) for R(z)>0. Show that h is harmonic and that h(z) is greater than 0 and less than pi d)Show that log(z-ic) is defined for R(z)>0 and any real number c. Prove that |F(log(z-ic))|is less than pi/2 in this region e) Prove that h(z)=F(log(z-ia) - log(z-ib)) f) Use the fundamental theorem... click for more