Complex exponential

The complex exponential extends the idea of exponentiation to complex numbers. For a real number $x$, $e^x$ is the exponential function. For a complex number $z = x + iy$, we define $e^z$ using Euler's formula: $e^{x+iy} = e^x ( ext{cos } y + i ext{sin } y).$ This means the exponential of a complex number combines both growth (from $e^x$) and rotation (from $ext{cos } y + i ext{sin } y$) in the complex plane.

Important properties

• $e^{z_1 + z_2} = e^{z_1} e^{z_2}$ for any complex numbers $z_1, z_2$

• Euler's formula: $e^{iy} = ext{cos } y + i ext{sin } y$

• $|e^{iy}| = 1$ for any real $y$ (lies on the unit circle)

• The exponential function is periodic in the imaginary part: $e^{i(y + 2\pi)} = e^{iy}$