AcademyComplex Form
Academy
Complex Exponential
Level 1 - Math I (Physics) topic page in Complex Form.
Complex Exponentials
The exponential function extends naturally to complex arguments, providing powerful tools for complex analysis.
Definition
For any complex number \(z = x + iy\):
Complex Exponential
\[e^z = e^{x+iy} = e^x(\cos y + i\sin y)\]
This definition preserves the key property \(e^{z_1 + z_2} = e^{z_1} \cdot e^{z_2}\).
Properties
The complex exponential shares fundamental properties with its real counterpart:
Product Rule
\[e^{z_1 + z_2} = e^{z_1} e^{z_2}\]
Quotient Rule
\[\frac{e^{z_1}}{e^{z_2}} = e^{z_1 - z_2}\]
Periodicity
\[e^{z + 2\pi i} = e^z\]
The complex exponential is periodic with period \(2\pi i\).
Relationship to De Moivre's Theorem
Using Euler's formula, we can derive De Moivre's theorem:
From Euler
\[(e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)\]
This unified view connects:
- Complex exponentials
- Trigonometric functions
- Complex multiplication (rotation and scaling)
Complex Powers
For any complex base \(a\) and exponent \(b\):
Complex Power
\[a^b = e^{b \ln a}\]
where \(\ln a\) is the complex logarithm, having infinitely many values due to the periodicity of the exponential.