Complex numbers are really complex...

Question

Show that tan(iz)=i tanh z

Evaluate the values of following expresiions; giving answers in the form of x+iy please.

squareroot(1+i)

exp(Log(3+4i))

Answer 1

1.) Use trigonometric identities then convert to exponential form. i.e. :

tan(iz) = sin(iz)/cos(iz) = [(e^i(iz) - e^-i(iz))/2i]/[(e^i(iz) + e^-i(iz))/2] = [i(e^z - e^-z)/2]/[(e^-z + e^z)/2]

= i sinh(z) / cosh(z) = i tanh(z)

2.) Convert to exponential form. We have:

sqrt(1 + i) = (sqrt(2)e^iπ/4)^1/2 = 2^1/4e^iπ/8 = 2^1/4[cos(π/8) + isin(π/8)]

3.) exp(log(z)) = z, so the answer is just 3 + 4i.