Solve for locus of points in the complex plane and please describe all sol'n steps

|z|= |z-i| z = x + yi

Find locus of all points in the complex plane.

Please give details of soln |z-i| + |z| = 9

Determine expressions from a given formulae

I have the answers to this question, although I would like to know how to get to answers from the given formulae. Please list all steps in as much detail as possible. thanks

Root test

a) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity" ps. {An} is a sequence of non-zero complex numbers b) Prove that although the following power series have R=1 sum(nz^n) does not converge on any point of the unit circle

Power series - holomorphic function

b) Show that for |z|<1 we have (z/1-z^2)+(z^2/1-z^4)+.......(z^2n/1-z^2n+1)=(z/1-z) and (z/1+z)+(2z^2/1+z^2)+....(2^kz^2k/1+z^2k)=(z/1-z) ps. maybe dyadic expansion of an integer may be used here or the fact that 1+2+2^2....+2^k=2^k+1 -1 c) Find the holomorphic function of z that vanishes at z=0 and has real part u( ...continues

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 mi ...continues

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_ ...continues

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 ...continues

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 ...continues