Show that the function f z sinz maps the yaxis onetoone to

Show that the function f (z) = sin(z) maps the y-axis one-to-one to the imaginary axis.

Solution

One way to proceed is to note that

sin(z)=(e^ize^iz )/2i.

So

sin(iy)=(e^ye^y)/2i.

The function f(y)=e^ye^y takes R to R.

The derivative

f(y)=ye^ye^y=ye^y(1e^2y) is always negative

So f(y) is injective. As lim y± implies f(y)=, we see that ff is surjective.

Thus sin(iy)=i2f(y)sin(iy)=i2f(y) takes every value on the imaginary axis exactly once.