Write \(e^{x+iy}\) in terms of real functions of \(x\) and \(y\).

What is \(|e^{x+iy}|\)?

Find \(e^{i\pi}\).

Find \(|e^{3+2i}|\).

Find the real and imaginary parts of \(e^{1+i\pi/2}\).

Evaluate \(\cos(iu)\) using the exponential definition.

Evaluate \(\sin(iu)\) using the exponential definition.

Show that \(e^{z+2\pi i}=e^z\).

Find \(e^{(1+i)(2-i)}\) in the form \(e^x(\cos y+i\sin y)\).

Find \(\operatorname{Re}(e^{2+i\pi})\) and \(\operatorname{Im}(e^{2+i\pi})\).

Explain why \(e^z\) never equals \(0\).

Verify the identity \(\cosh(iz)=\cos z\).

Find all \(z=x+iy\) such that \(|e^z|=e^4\).

Find all \(z=x+iy\) such that \(e^z\) is a positive real number.

For which real \(y\) is \(e^{2+iy}\) purely imaginary?

Find all real \(x,y\) such that \(e^{x+iy}=e^2\).

Show that \(\sinh(iz)=i\sin z\).

A learner claims \(\arg(e^{x+iy})=y\) exactly. Diagnose the statement.

Prove that \(e^z\) is periodic in the imaginary direction with period \(2\pi i\), but not with real period \(2\pi\).

Derive \(\cos^2 z+\sin^2 z=1\) from the exponential definitions.