Weierstrass substitution
E110609
The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weierstrass substitution canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T940263 — 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: Weierstrass substitution Context triple: [Karl Weierstrass, notableFor, Weierstrass substitution]
-
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.
Peuterey Integral
Peuterey Integral is a legendary, highly committing alpine climbing route that follows the full Peuterey ridge to the summit of Mont Blanc, renowned as one of the longest and most serious ridge climbs in the Alps.
-
C.
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.
-
D.
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.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- 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: Weierstrass substitution Target entity description: The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
-
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.
Peuterey Integral
Peuterey Integral is a legendary, highly committing alpine climbing route that follows the full Peuterey ridge to the summit of Mont Blanc, renowned as one of the longest and most serious ridge climbs in the Alps.
-
C.
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.
-
D.
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.
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
integration technique
ⓘ
trigonometric substitution method ⓘ |
| advantage |
provides a systematic method for many trigonometric integrals
ⓘ
reduces trigonometric integrals to algebraic integrals ⓘ |
| alsoKnownAs |
tangent half-angle substitution
ⓘ
universal trigonometric substitution ⓘ |
| appliesTo |
definite integrals
ⓘ
indefinite integrals ⓘ integrals of rational functions of sin(x) and cos(x) ⓘ integrals of rational functions of tan(x) and sec(x) ⓘ |
| basedOnSubstitution | t = tan(x/2) ⓘ |
| category | techniques of integration ⓘ |
| disadvantage | can lead to algebraically complicated expressions ⓘ |
| domainCondition | t = tan(x/2) is defined where cos(x/2) ≠ 0 ⓘ |
| field |
calculus
ⓘ
mathematical analysis ⓘ |
| generalizationOf | basic trigonometric substitutions in integration ⓘ |
| hasProperty | universal for rational functions of sin(x) and cos(x) without radicals ⓘ |
| historicalAttribution | 19th-century analysis ⓘ |
| implies |
cos(x) = (1 - t^2)/(1 + t^2)
ⓘ
dx = 2 dt/(1 + t^2) ⓘ sin(x) = 2t/(1 + t^2) ⓘ |
| mapsTo | rational functions in t ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| notation | t = tan(x/2) is often denoted by t = tan(θ/2) or u = tan(x/2) ⓘ |
| oftenContrastedWith | standard right-triangle trigonometric substitutions ⓘ |
| relatedTo |
Euler substitution
ⓘ
partial fraction decomposition ⓘ substitution rule in integration ⓘ trigonometric identities ⓘ |
| requires |
knowledge of half-angle formulas
ⓘ
knowledge of tangent function ⓘ |
| requiresCareWith | back-substitution and determination of correct angle ⓘ |
| typicalForm | ∫R(sin x, cos x) dx → ∫R(2t/(1+t^2), (1−t^2)/(1+t^2))·2 dt/(1+t^2) ⓘ |
| usedFor |
evaluating integrals involving sine and cosine
ⓘ
simplifying integration of rational functions of sin(x) and cos(x) ⓘ transforming trigonometric integrals into rational integrals ⓘ |
| usedIn |
computer algebra systems
ⓘ
symbolic integration ⓘ |
| usedInEducationLevel |
advanced high school mathematics
ⓘ
undergraduate calculus ⓘ |
| worksBy | expressing sine and cosine as rational functions of tan(x/2) ⓘ |
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
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Input
Subject: Weierstrass substitution Description of subject: The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.