An integral transform mapping a function f(r) on the positive half-line to F(s) = ∫_0^∞ r^{s-1} f(r) dr, converting dilation (scaling) behavior near r = 0 or ∞ into meromorphic dependence on the complex parameter s; extensively used to analyze singularities and boundary asymptotics, especially near conical points.

Principle

Under the Mellin transform, scaling r → λ r corresponds to translation s → s + log_λ in the complex parameter; singular power-law behaviors r^α produce poles or residues at s = −α, so asymptotic expansions in r map to discrete spectral data in s.

Demonstration

To analyze a PDE on a cone near the vertex, apply the Mellin transform in the radial variable to convert radial derivatives into algebraic operations in s and obtain a family of parameter-dependent ordinary or pseudodifferential equations on the link; the location and order of poles of the transformed system determine possible singular exponents in the radial expansion.

Misapplication

Applying the Mellin transform to functions without appropriate decay or analytic continuation in a vertical strip of s; failure to verify integrability or strip of holomorphy leads to incorrect pole identification and mischaracterization of asymptotics.

Consequence

The Mellin transform converts local scaling problems into spectral parameter problems: poles and residues of the transformed family give explicit coefficients and exponents of asymptotic expansions, enabling precise singularity and index computations for boundary and corner problems.

Reversal

The Fourier transform is the dual picture for translations rather than dilations: while Fourier converts shifts into phase multipliers and studies translational invariance, Mellin specializes to scale invariance and radial asymptotics.

Boundary

Valid for functions (or distributions) on (0, ∞) with growth/decay guaranteeing convergence in a vertical strip of complex s and for problems exhibiting dilation invariance or local conical geometry; not directly applicable to problems lacking a radial/dilation variable or where no appropriate strip of holomorphy exists.

Semantic Tension

Tension exists with Laplace/Fourier and z-transforms: all are integral transforms encoding different symmetries (translations, dilations, discrete shifts) and can sometimes be interconverted under change of variables, but the Mellin transform is uniquely suited to scaling and power-law asymptotics.

Synthesis

The Mellin transform is the tool that translates radial scaling and power-law singularities into meromorphic parameter families in s; by mapping dilation behavior to pole structure it provides a precise calculus for boundary asymptotics, index computations, and analysis near conical or conical-type singularities.