Mellin transforms
E637297
Mellin transforms are integral transforms that convert functions into complex-variable representations, playing a central role in analytic number theory by linking arithmetic functions to Dirichlet series and zeta functions.
All labels observed (5)
How this entity was disambiguated
This entity first appeared as the object of triple T7030747 — 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: Mellin transforms Context triple: [Multiplicative Number Theory, hasKeyTool, Mellin transforms]
-
A.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
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B.
Sommerfeld-Watson transform
The Sommerfeld-Watson transform is a complex-analysis technique that converts discrete sums over angular momentum into contour integrals, widely used in scattering theory and Regge theory to study analytic properties of amplitudes.
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C.
Laplace transform
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
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D.
Tauberian theorems
Tauberian theorems are results in mathematical analysis that connect the behavior of transformed series or integrals (such as those summed by Abel or Cesàro methods) back to the asymptotic behavior or convergence of the original sequences or series.
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E.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
- 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: Mellin transforms Target entity description: Mellin transforms are integral transforms that convert functions into complex-variable representations, playing a central role in analytic number theory by linking arithmetic functions to Dirichlet series and zeta functions.
-
A.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
B.
Sommerfeld-Watson transform
The Sommerfeld-Watson transform is a complex-analysis technique that converts discrete sums over angular momentum into contour integrals, widely used in scattering theory and Regge theory to study analytic properties of amplitudes.
-
C.
Laplace transform
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
-
D.
Tauberian theorems
Tauberian theorems are results in mathematical analysis that connect the behavior of transformed series or integrals (such as those summed by Abel or Cesàro methods) back to the asymptotic behavior or convergence of the original sequences or series.
-
E.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
integral transform
ⓘ
mathematical concept ⓘ |
| appliedIn |
analytic continuation of Dirichlet series
ⓘ
conformal field theory ⓘ image processing ⓘ modular forms ⓘ pattern recognition ⓘ physics ⓘ probability theory ⓘ quantum field theory ⓘ statistics ⓘ study of the Riemann zeta function ⓘ |
| convergenceDependsOn | growth of f(x) near 0 and ∞ ⓘ |
| definition |
For a function f(x), the Mellin transform is M{f}(s) = ∫₀^∞ x^{s-1} f(x) dx when the integral converges
ⓘ
linked to:
Mellin transforms
|
| field |
analysis
ⓘ
analytic number theory ⓘ complex analysis ⓘ harmonic analysis ⓘ |
| hasVariant |
discrete Mellin transform
ⓘ
fractional Mellin transform ⓘ two-dimensional Mellin transform ⓘ
linked to:
Mellin transforms
|
| historicalPeriod | late 19th century ⓘ |
| inverseTransform | f(x) = (1/2πi) ∫_{c-i∞}^{c+i∞} x^{-s} F(s) ds under suitable conditions ⓘ |
| kernel | x^{s-1} ⓘ |
| mapsFrom | functions on positive real numbers ⓘ |
| mapsTo | functions of a complex variable ⓘ |
| namedAfter | Hjalmar Mellin ⓘ |
| property |
admits Parseval-type identities
ⓘ
converts multiplicative convolution into ordinary products ⓘ has an inversion formula under suitable analytic conditions ⓘ turns scaling in the original variable into translation in the transform variable ⓘ |
| relatedTo |
Fourier transform
ⓘ
Laplace transform ⓘ bilateral Laplace transform ⓘ |
| typicalCondition |
f(x) locally integrable on (0,∞)
ⓘ
s lies in a vertical strip of convergence in the complex plane ⓘ |
| usedFor |
analysis of algorithms
ⓘ
evaluation of integrals ⓘ linking arithmetic functions to Dirichlet series ⓘ linking arithmetic functions to zeta functions ⓘ representation of functions as complex-variable functions ⓘ scale-invariant signal analysis ⓘ solution of differential equations ⓘ study of asymptotic expansions ⓘ |
| usedWith |
Dirichlet L-functions
ⓘ
Gamma function ⓘ Riemann zeta function ⓘ |
| variableOfTransform | complex variable s ⓘ |
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: Mellin transforms Description of subject: Mellin transforms are integral transforms that convert functions into complex-variable representations, playing a central role in analytic number theory by linking arithmetic functions to Dirichlet series and zeta functions.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Mellin transforms
linked to: Mellin transforms
linked to: Mellin transforms
linked to: Mellin transforms
linked to: Mellin transforms
Mellin transform
→
definition
→
For a function f(x), the Mellin transform is M{f}(s) = ∫₀^∞ x^{s-1} f(x) dx when the integral converges
ⓘ
subject linked to:
Mellin transforms
linked to: Mellin transforms
subject linked to:
Mellin transforms
linked to: Mellin transforms