# Complex maximum principle application over absolute value of complex sine

I'm new into using the complex maximum principle so I'm stuck calculating the maximum of the function $f(z)$= $|\sin z|$ over the set $[0,1] \times [-1,1]$.

In which points or point the maximum holds. My line of thought goes as the following:

We can write

$$\sin(z) = \sin(x+iy) = \sin x \cos iy + \sin iy \cos x = \sin x \cosh y + i \sinh y \cos x.$$

Then we calculate the square norm so we get:

$$|\sin(z)|^2 = \sin^2 x \cosh^2 y + i \sinh^2 y \cos^2 x$$

And I know that because the complex maximum principle the supremum of $|\sin(z)|$ is attained on the boundary of the rectangle $[0,1] \times [-1,1]$.

Can someone please help me filling up the gaps in the calculations and proof so I can understand how to properly use the maximum principle over this function?? Thanks

• Is sen meant to be $\sin$? – copper.hat Nov 8 '15 at 21:38
• Yes, I was thinking in spanish (sine-seno then sin-sen). Sorry – Cervus Nov 8 '15 at 21:44
• In your formula for $\lvert \sin (z)\rvert^2$, you have an $i$ that doesn't belong there. $\lvert \sin z\rvert^2$ is of course real, so the correct expression is $\lvert \sin z\rvert^2 = \sin^2 x \cosh^2 y + \sinh^2 y\cos^2 x$. Now use $\cosh^2 y = 1 + \sinh^2 y$, and $\sin^2 x + \cos^2 x = 1$ to get another expression for $\lvert\sin z\rvert^2$ that makes it obvious at which points of the rectangle $\lvert \sin z\rvert$ attains its maximum. – Daniel Fischer Nov 8 '15 at 22:23