Functions of Complex Variable

Question

if 

tan(x+i*y)=A+i*B

Show that 

A/B=sin(2*x)/sinh(2*y)

Answer 1

First you have to separate real and imaginary parts of Tan(x+iy)=Tan(z)=sin(z)/cos(z)

 

sinz=sin(x+iy)=sinxcos(iy)+cosxsin(iy)=sinxcoshy-icosx sinhy

cosz=cos(x+iy)=cosxcos(iy)-sinxsin(iy)=cosxcoshy−isinxsinhy

 

Now if you plug in Tan(z) and simplify (it is easy!) you get

 

Tan(z)=(sin(2x)+isinh(2y))/(cos(2x)+cosh(2y))= A+iB.

This means that 

 

A=sin(2x)/(cos(2x)+cosh(2y)) and B= sinh(2y)/(cos(2x)+cosh(2y))

 

Now,

 A/B=sin(2x)/sinh(2y)

 

If any questions, let me know.