State the definition of \(e^{x+iy}\) for real \(x\) and \(y\).

What is the period of the complex exponential in the imaginary direction?

Evaluate \(e^{i\pi/2}\).

Evaluate \(e^{2+i0}\).

Write \(e^{1+i\pi}\) in Cartesian form.

Find the modulus of \(e^{3-2i}\).

Find the principal argument of \(e^{-1+i5\pi/6}\).

Simplify \(e^{2+i\pi/3}e^{-1+i\pi/6}\).

Solve \(e^z=1\) for complex \(z\).

Solve \(e^z=-1\) for complex \(z\).

Explain why \(e^{z_1+z_2}=e^{z_1}e^{z_2}\) is consistent with rotation and scaling.

A student says \(e^{i\theta}\) grows as \(\theta\) grows because exponentials grow. Correct the statement.

Find all \(z\) such that \(e^z=2i\).

Find all \(z\) such that \(e^{2z}=1\).

For which real \(t\) is \(|e^{t+i\pi/4}|=5\)?

For which real \(t\) is \(e^{1+it}\) purely imaginary and above the origin?

For which real \(t\) does \(e^{t+i t}\) lie on the unit circle?

Prove that \(e^{z+2\pi i}=e^z\) for every complex \(z\).

Show that \(e^z\ne0\) for every complex \(z\).

Explain why the equation \(e^z=w\) has infinitely many solutions for every non-zero complex number \(w\).