If `z=r e^(itheta)` , then prove that `|e^(i z)|=e^(-r sintheta)`

If |z-i R e(z)|=|z-I m(z)| , then prove that z , lies on the bisectors of the quadrants.

If z!=0 is a complex number, then prove that R e(z)=0 rArr Im(z^2)=0.

If z=z_0+A( bar z -( bar z _0)), w h e r eA is a constant, then prove that locus of z is a straight line.

Consider an ellipse having its foci at A(z_1)a n dB(z_2) in the Argand plane. If the eccentricity of the ellipse be e and it is known that origin is an interior point of the ellipse, then prove that e in (0,(|z_1-z_2|)/(|z_1|+|z_2|))

Write the following in z=a+ib form: z=e^(-itheta)

Write the following in z=a+ib form: z=e^(a+itheta)

Let alpha+ibeta be a complex number and let x+iy=e^(alpha+ibeta) where (x,y,alpha,beta in R) then we define log_e(x+iy)=alpha+ibeta x+iy=re^(itheta) ,where r=sqrt(x^2+y^2),theta=tan^(-1)(y/x), alpha+ibeta=log_e(x+iy)=log_e(r.e^(itheta)=log_er+itheta so log_e(x+iy)=log_er+itheta log_e(z)=log_eabsz+iarg(z) log_e(-i) equals

The three angular points of a triangle are given by Z=alpha, Z=beta,Z=gamma,w h e r ealpha,beta,gamma are complex numbers, then prove that the perpendicular from the angular point Z=alpha to the opposite side is given by the equation R e((Z-alpha)/(beta-gamma))=0

For any complex number z prove that |R e(z)|+|I m(z)|<=sqrt(2)|z|

Let alpha+ibeta be a complex number and let x+iy=e^(alpha+ibeta) where (x,y,alpha,beta in R) then we define log_e(x+iy)=alpha+ibeta x+iy=re^(itheta) ,where r=sqrt(x^2+y^2),theta=tan^(-1)(y/x), alpha+ibeta=log_e(x+iy)=log_e(r.e^(itheta)=log_er+itheta so log_e(x+iy)=log_er+itheta log_e(z)=log_eabsz+iarg(z) if sin(log_ei)=a+ib then value of a and b is