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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 non-zero 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. Therefore \(a^b\) is generally multi-valued unless a branch of \(\ln\) is chosen.

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De Moivre Theorem

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Euler Formula