Mellin transforms

E637297

integral transform mathematical concept

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.

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Observed surface forms (2)

Surface form	Occurrences
Mellin transform	1
Mellin convolution	1

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 NERFINISHED ⓘ
field	analysis ⓘ analytic number theory ⓘ complex analysis ⓘ harmonic analysis ⓘ
hasVariant	discrete Mellin transform ⓘ fractional Mellin transform ⓘ two-dimensional Mellin transform NERFINISHED ⓘ
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 NERFINISHED ⓘ
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 NERFINISHED ⓘ Laplace transform NERFINISHED ⓘ 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 NERFINISHED ⓘ Gamma function NERFINISHED ⓘ Riemann zeta function NERFINISHED ⓘ
variableOfTransform	complex variable s ⓘ

Referenced by (3)

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

Multiplicative Number Theory → hasKeyTool → Mellin transforms ⓘ

Hadamard fractional integral → connectedTo → Mellin transforms ⓘ

this entity surface form: Mellin convolution

Dirichlet series → relatedTo → Mellin transforms ⓘ

this entity surface form: Mellin transform