Euler's Formula In Complex Analysis

Euler’s formula in complex analysis

Euler’s formula is an important result in complex analysis, expressing the connection between exponentials and trigonometric functions. It states that, for any real \theta:

e^{i\theta} = \cos(\theta) + i\sin(\theta)

This result can be derived from the Maclaurin series for the exponential, cosine, and sine functions.

Substituting z = i\theta into the power series for e^z, and grouping real and imaginary terms, gives the series for \cos(\theta) and \sin(\theta). This also shows how analytic functions of complex variables connect with real functions.

Geometrically, e^{i\theta} represents a unit complex number on the complex plane, tracing the unit circle as \theta varies. Euler’s formula therefore serves as the basis for representing any complex number z in polar form:

z = r e^{i\theta}

where r is the modulus and \theta the argument of z. The conjugate is simply r e^{-i\theta}, making the polar representation convenient for multiplication, division, and exponentiation.

A direct consequence is De Moivre’s formula, describing powers of complex numbers:

(\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)

This follows from Euler’s formula using exponent laws. Such properties underlie many results in both mathematics and physics, from signal analysis to quantum mechanics.

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