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Functions of Complex Variable

if 
tan(x+i*y)=A+i*B
Show that 
A/B=sin(2*x)/sinh(2*y)

Comments

It seems to me I made some typo mistakes, but the approach (idea) is correct. So if you look carefully,you can find mistakes and get the right answer.
 
There is also another way, too: 
Tan(z)=(1/i)(eiz-e-iz)/(eiz+e-iz). From this expression you can separate real and imaginary parts.
 

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Bobosharif S. | Mathematics/Statistics TutorMathematics/Statistics Tutor
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.

Comments

Ok
 We have Tan(x+iy) and we need to separate real and imaginary parts: A and B (as you denoted). How?
 
We know that
1. Tan(x+iy)=sin(x+iy)/cos(x+iy).
 
Now we have to remind a few identities (formulas):
a) sin(a+b)=...
b) cos(a+b)=...
c) cosh(a)=cos(ia)
d) sinh(a)=-isin(ia)
 
2. Write down (expand) tan(x+iy)= .. and by using a)-d) simplify that such that you can see real and imaginary parts. That all.