Show that the relation fx iyI fx fiy is satisfied byfz si

Show that the relation |f(x + iy)I = |f(x) +f(iy)| is satisfied byf(z) = sin z. Can you find any other functions for which it is true?

Solution

f(z)=sinz

plug z=x+iy

=> f(x+iy) = sin(x+iy)

using sin(a+B) = sinAcosB + cosAsinB

f(x+iy) = sinxcos(iy) + cosxsin(iy)

now |f(x+iy)| = |sinxcos(iy) + cosxsin(iy)|

                   = |sinxcosh(y) + isinh(y)cosx|     ------> as cos(iy)= cosh(y) & sinh(y) = isinh(y)

                   = |sinx + isinh(y)|   -----------(1)

and

|f(x) + f(iy)| = |sinx + sin(iy)| = |sinx + isinh(y)| ----------(2)

(1) = (2)

hence proved