Ken T.

asked • 01/13/18

Functions of Complex Variable

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

Bobosharif S.

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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01/13/18

1 Expert Answer

By:

Bobosharif S. answered • 01/13/18

Tutor
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Mathematics/Statistics Tutor

Ken T.

Could you please  explain more step by step
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01/14/18

Bobosharif S.

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.
 
 
 
 
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01/14/18

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