Locus of points that satisfy the equation of a line, circle, hyperbola, parabola or ellipse.

In the following problems determine the set of Z satisfying the given equations 1. |Z|=|Z-i| 2. |Z|^2 + Im(z)= 16 3. |Z-i|+|Z| = 9 Where: Z = x+yi __ Z = x-yi

Solving for the locus of points in the complex plane

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

Finding the locus of all points in the complex plane.

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

Writing Equations from Formulas

An equation derived from an a.c. bridge circuit is given by Components R3,R4,C1 and C4 have known values. Determine expressions for Rx and Cx in terms of the known components. keywords: formulae

Limits : Proving the Root Test Assuming the Ratio 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 and 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

Holomorphisms

Suppose that f is holomorphic 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

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

Finding an Integral for the Positive Sense of a Circle

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 p ...continues

Fractional Transformations, Cross Ratios and 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|