Euler’s reflection formula

E596522

mathematical identity

Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.

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All labels observed (2)

Label	Occurrences
Gamma function reflection formula	2
Euler’s reflection formula canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T6482564 — 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 reflection formula
Context triple: [Gamma function, relatedFunction, Euler’s reflection formula]

A. 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.
B. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
C. 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.
D. Euler’s formula for complex exponentials
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.
E. Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
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 reflection formula
Target entity description: Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.

A. 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.
B. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
C. 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.
D. Euler’s formula for complex exponentials
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.
E. Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
F. None of above. chosen

Statements (31)

Predicate	Object
instanceOf	mathematical identity ⓘ
appearsIn	analytic number theory ⓘ complex analysis textbooks ⓘ theory of special functions ⓘ
classification	reflection formula for special functions ⓘ
connects	Gamma function and trigonometric functions ⓘ
domainCondition	z not an integer ⓘ z ∉ ℤ ⓘ
field	complex analysis ⓘ special functions ⓘ
hasAlternativeForm	Γ(z) Γ(1 - z) = π csc(πz) ⓘ Γ(z) Γ(1 - z) sin(πz) = π ⓘ
hasExpression	Γ(z) Γ(1 - z) = π / sin(πz) ⓘ
implies	zeros of sin(πz) correspond to poles of Γ(z) Γ(1 - z) ⓘ Γ(z) Γ(1 - z) is meromorphic with simple poles at integers ⓘ
involvesFunction	Gamma function NERFINISHED ⓘ sine function ⓘ
namedAfter	Leonhard Euler NERFINISHED ⓘ
relatedTo	Euler’s formula for the sine function via infinite product NERFINISHED ⓘ Gamma function functional equation Γ(z+1) = z Γ(z) ⓘ Riemann zeta function functional equations ⓘ
relates	sin(πz) ⓘ Γ(1 - z) ⓘ Γ(z) ⓘ
revealsPropertyOf	symmetry of the Gamma function ⓘ
typeOfSymmetry	reflection across the line Re(z) = 1/2 ⓘ
usedFor	analytic continuation of the Gamma function ⓘ deriving identities for trigonometric functions ⓘ relating Γ(z) and Γ(1 - z) ⓘ studying functional equations of special functions ⓘ
validOn	complex plane minus the integers ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Gamma function → relatedFunction → Euler’s reflection formula ⓘ

Selberg integral → involves → Euler’s reflection formula ⓘ

this entity surface form: Gamma function reflection formula

Stirling's approximation → relatedTo → Euler’s reflection formula ⓘ