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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| De Moivre's formula | 1 |
| Euler's identity | 1 |
| Euler’s formula for complex exponentials canonical | 1 |
| Euler’s identity e^{iπ}+1=0 | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T426763 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Euler’s formula for complex exponentials Context triple: [Leonhard Euler, notableWork, Euler’s formula for complex exponentials]
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A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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B.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
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C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
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E.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Euler’s formula for complex exponentials
Target entity description: 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.
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
E.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Euler’s formula for complex exponentials
Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Euler’s identity e^{iπ}+1=0
this entity surface form:
De Moivre's formula
subject surface form:
Leonhard Euler
this entity surface form:
Euler's identity