Euler’s formula for complex exponentials
E54268
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
Observed surface forms (3)
| Surface form | Occurrences |
|---|---|
| De Moivre's formula | 1 |
| Euler's identity | 1 |
| Euler’s identity e^{iπ}+1=0 | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
formula in complex analysis
ⓘ
mathematical identity ⓘ trigonometric identity ⓘ |
| associatedWith | Leonhard Euler ⓘ |
| canBeExtendedTo | complex \theta ⓘ |
| category | identity involving transcendental functions ⓘ |
| codomain | unit circle in \mathbb{C} ⓘ |
| derivedFrom |
power series of \cos z
ⓘ
power series of \sin z ⓘ power series of e^{z} ⓘ |
| domainOfVariable | \theta \in \mathbb{R} ⓘ |
| equivalentForm |
\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}
ⓘ
\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} ⓘ |
| geometricInterpretation | e^{i\theta} represents a rotation by angle \theta in the complex plane ⓘ |
| hasExpression | e^{i\theta} = \cos\theta + i\sin\theta ⓘ |
| hasMagnitudeProperty | |e^{i\theta}| = 1 ⓘ |
| hasSpecialCase | Euler’s identity e^{i\pi} + 1 = 0 ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| implies |
Im(e^{i\theta}) = \sin\theta
ⓘ
Re(e^{i\theta}) = \cos\theta ⓘ e^{i(\theta + 2\pi)} = e^{i\theta} ⓘ periodicity of complex exponential on the imaginary axis ⓘ |
| involvesConstant | imaginary unit i ⓘ |
| involvesFunction |
complex exponential function
ⓘ
cosine function ⓘ sine function ⓘ |
| involvesVariable | real angle \theta ⓘ |
| relatesConcept |
complex numbers
ⓘ
polar representation of complex numbers ⓘ trigonometric functions ⓘ unit circle in the complex plane ⓘ |
| underpins |
complex analysis
ⓘ
engineering mathematics ⓘ theory of rotations in the plane ⓘ |
| usedFor |
AC circuit analysis
ⓘ
Fourier analysis ⓘ
surface form:
Fourier series
Fourier analysis ⓘ
surface form:
Fourier transforms
converting between exponential and trigonometric forms ⓘ deriving Euler’s identity e^{i\pi} + 1 = 0 ⓘ expressing complex numbers in polar form ⓘ phasor analysis in electrical engineering ⓘ representing sinusoidal oscillations as complex exponentials ⓘ signal processing ⓘ solving linear differential equations with constant coefficients ⓘ |
| usedIn |
control theory
ⓘ
quantum mechanics ⓘ vibrations and waves analysis ⓘ |
| validFor | all real \theta ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
De Moivre's formula
subject surface form:
Leonhard Euler
this entity surface form:
Euler's identity
this entity surface form:
Euler’s identity e^{iπ}+1=0