# Mellin transform

The Mellin transform is an integral transform that converts a function defined on the positive real axis into a function of a complex variable, defined by

\[ \mathcal{M}f(z) = \int_{0}^{\infty} t^{z-1} f(t)\,dt \]

whenever the integral converges. It is the multiplicative counterpart of the [Fourier transform](https://www.edgepedia.org/fourier-transform): where [Fourier analysis](https://www.edgepedia.org/fourier-analysis) decomposes a signal under translations, the Mellin transform decomposes it under dilations, and it is the natural tool for scale-invariant problems.<sup>[1](https://dlmf.nist.gov/2.5)</sup><sup> • </sup><sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S1051200424005049)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \mathcal{M}f(z) = \int_{0}^{\infty} t^{z-1} f(t)\,dt \) for locally integrable \( f \) on \( (0,\infty) \)<sup>[1](https://dlmf.nist.gov/2.5)</sup> |
| Domain of analyticity | Usually an infinite strip \( a < \Re z < b \) parallel to the imaginary axis, called the fundamental strip<sup>[1](https://dlmf.nist.gov/2.5)</sup><sup> • </sup><sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> |
| Inversion | \( f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-z} \mathcal{M}f(z)\,dz \), with \( a < c < b \)<sup>[1](https://dlmf.nist.gov/2.5)</sup> |
| Scaling behavior | If \( g(x) = f(\lambda x) \) then \( g^{*}(s) = \lambda^{-s} f^{*}(s) \), so dilation becomes multiplication by an exponential<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> |
| Convolution | The Mellin convolution \( I(x) = \int_{0}^{\infty} f(t)h(xt)\,dt \) satisfies \( \mathcal{M}I(z) = \mathcal{M}f(1-z)\,\mathcal{M}h(z) \)<sup>[1](https://dlmf.nist.gov/2.5)</sup> |
| Prototype example | The transform of \( e^{-x} \) is the gamma function \( \Gamma(s) \), with fundamental strip \( \langle 0, +\infty \rangle \)<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> |
| Number-theory role | The Mellin transform of the theta function gives Riemann's proof of the functional equation of \( \zeta(s) \); for modular forms it produces \( (2\pi)^{-s}\Gamma(s)L_f(s) \)<sup>[4](https://people.math.harvard.edu/~elkies/M259.06/zeta1.pdf)</sup><sup> • </sup><sup>[5](https://uwaterloo.ca/pure-mathematics/sites/default/files/uploads/documents/pmath_944_lecture_notes_-_modular_forms.pdf)</sup> |

## Definition and basic properties

Let \( f(t) \) be locally integrable on \( (0,\infty) \), meaning that \( \int_{\rho}^{T} f(t)\,dt \) exists for all \( 0 < \rho < T < \infty \). The Mellin transform is

\[ \mathcal{M}f(z) = \int_{0}^{\infty} t^{z-1} f(t)\,dt, \]

when this integral converges.<sup>[1](https://dlmf.nist.gov/2.5)</sup> The domain of analyticity is usually an infinite strip \( a < \Re z < b \) parallel to the imaginary axis; in the asymptotic-analysis literature this is called the fundamental strip. Outside the strip the transform is defined by analytic continuation, and in general it is analytic in the strip and meromorphic outside.<sup>[1](https://dlmf.nist.gov/2.5)</sup><sup> • </sup><sup>[6](https://escholarship.org/content/qt8z61b9g5/qt8z61b9g5_noSplash_f53bec10912b7dddf2fb5abc06c09064.pdf?t=rcwcwe)</sup>

Convergence at the two endpoints is controlled by the behavior of \( f \) near 0 and \( \infty \). If \( f(x) \) is continuous and is \( O(x^{-\alpha}) \) as \( x \to 0 \) and \( O(x^{-\beta}) \) as \( x \to \infty \), the transform is defined in the fundamental strip \( \alpha < \Re(s) < \beta \).<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> The inversion formula recovers \( f \) from a vertical contour integral inside the strip:

\[ f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-z} \mathcal{M}f(z)\,dz, \qquad a < c < b. \]

This is, in substance, the Fourier inversion formula for the multiplicative group \( \mathbb{R}_{+} \) with invariant measure \( dx/x \): writing \( x = e^{u} \) turns the Mellin transform into a Fourier transform in the logarithmic variable, which is why dilation of \( x \) corresponds to a simple exponential factor in \( s \).<sup>[1](https://dlmf.nist.gov/2.5)</sup><sup> • </sup><sup>[7](https://qft.org/mathematical-methods/asymptotic-special-functions/mellin-transforms-scaling-asymptotics/)</sup>

The transform has a convolution structure adapted to the multiplicative group. One of the two standard Mellin convolutions is

\[ I(x) = \int_{0}^{\infty} f(t)\,h(xt)\,dt, \qquad x > 0, \]

and its transform factors as \( \mathcal{M}I(z) = \mathcal{M}f(1-z)\,\mathcal{M}h(z) \). When \( x = 1 \) this identity is a Parseval-type formula.<sup>[1](https://dlmf.nist.gov/2.5)</sup>

## Asymptotic analysis

The most widely used feature of the transform is a correspondence between the asymptotic expansion of \( f(x) \) at 0 (respectively at \( +\infty \)) and the singularities of \( f^{*}(s) \) in a left (respectively right) half-plane.<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> If \( f^{*}(s) \) decays toward \( \pm i\infty \) and has only polar singularities, closing the inversion contour and applying the residue theorem gives expansions of the form

\[ f(x) = \sum_{\beta} \operatorname{Res} f^{*}(s)\,x^{s}\big|_{s=\beta} + O(x^{d}), \]

with the sum taken over poles on the relevant side of the fundamental strip. A pole of order \( k \) at \( \beta \) contributes a term \( Q_{k-1}(\log x)\,x^{\beta} \), a polynomial in \( \log x \) of degree \( k-1 \) times a power.<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup>

The scaling property makes this effective for harmonic sums, sums of the form \( F(x) = \sum_{k} \lambda_{k} f(\mu_{k} x) \). Since \( g(x) = f(\lambda x) \) transforms to \( \lambda^{-s} f^{*}(s) \), the transform of such a sum factors into a generalized Dirichlet series \( \sum_{k} \lambda_{k} \mu_{k}^{-s} \) times \( f^{*}(s) \).<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> Under conditions on decay and pole structure, the asymptotic behavior of the sum is read off from the residues of this product, a result known as the Mellin asymptotic summation formula.<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup> This technique is a primary tool in the average-case analysis of algorithms, including tries, radix exchange sort, and hashing schemes.<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup>

The same residue mechanism underlies the Mellin transform method for integrals \( I(\lambda) = \int_{0}^{\infty} h(\lambda p) f(p)\,dp \) as \( \lambda \to \infty \): the Parseval formula writes \( I(\lambda) \) as a contour integral of \( \lambda^{-z} \mathcal{M}[h;z]\,\mathcal{M}[f;1-z] \), and shifting the contour to the right picks up residues that give the asymptotic expansion, with an error controlled by the shifted contour.<sup>[6](https://escholarship.org/content/qt8z61b9g5/qt8z61b9g5_noSplash_f53bec10912b7dddf2fb5abc06c09064.pdf?t=rcwcwe)</sup>

## Number theory: zeta functions and modular forms

The Mellin transform entered mathematics through analytic number theory. In his 1859 paper on the [prime-counting function](https://www.edgepedia.org/prime-counting-function), [Bernhard Riemann](https://www.edgepedia.org/bernhard-riemann) expressed the completed zeta function \( \xi(s) \) as a Mellin integral involving the theta function \( \theta(u) = \sum_{n=-\infty}^{\infty} e^{-\pi n^{2} u} \):

\[ 2\xi(s) = \int_{0}^{\infty} (\theta(u) - 1)\, u^{s/2}\, \frac{du}{u}, \qquad \Re s > 0. \]

Using the modular identity \( \theta(1/u) = u^{1/2}\theta(u) \), which follows from Poisson summation, the integral splits at \( u = 1 \) and becomes manifestly symmetric under \( s \leftrightarrow 1-s \), which proves both the analytic continuation and the functional equation of \( \xi \).<sup>[4](https://people.math.harvard.edu/~elkies/M259.06/zeta1.pdf)</sup> This argument launched the link between modular forms and \( L \)-functions that remains central in number theory.<sup>[8](https://sites.math.rutgers.edu/~sdmiller/gelbart-miller-zeta.pdf)</sup>

The same construction works for modular forms. For a cusp form (special modular form with Fourier expansion; links to L-functions) \( f(z) = \sum_{n=1}^{\infty} c(n) q^{n} \) of weight \( 2k \), the Mellin transform is

\[ \mathcal{M}(f)(s) = \int_{0}^{\infty} f(it)\, t^{s-1}\,dt = (2\pi)^{-s} \Gamma(s)\, L_{f}(s), \]

where \( L_{f}(s) = \sum_{n=1}^{\infty} c(n)/n^{s} \) is the associated [Dirichlet series](https://www.edgepedia.org/dirichlet-series).<sup>[5](https://uwaterloo.ca/pure-mathematics/sites/default/files/uploads/documents/pmath_944_lecture_notes_-_modular_forms.pdf)</sup> The modular transformation law of \( f \) under \( z \mapsto -1/z \) then converts into a functional equation for the completed \( L \)-function, giving analytic continuation to the whole complex plane; Hecke's converse theorem shows that this correspondence is two-way, so a Dirichlet series with the right functional equation comes from a modular form.<sup>[9](https://math.ou.edu/~kmartin/mfs/mfs.pdf)</sup> The prototype is \( f(z) = e^{iz} \), whose Mellin transform is \( \Gamma(s) \) for \( \Re s > 0 \).<sup>[5](https://uwaterloo.ca/pure-mathematics/sites/default/files/uploads/documents/pmath_944_lecture_notes_-_modular_forms.pdf)</sup>

The transform is named for the Finnish mathematician [Hjalmar Mellin](https://www.edgepedia.org/hjalmar-mellin), whose publications include "Ueber eine Verallgemeinerung der Riemann'schen Function zeta(s)" in Acta Societatis Scientiarum Fennicae 24 (1899) and further papers in Acta Mathematica in 1901.<sup>[10](https://mathshistory.st-andrews.ac.uk/Extras/Mellin_publications/)</sup>

## Scale invariance and applications

Just as the Fourier and Laplace transforms are the tools for linear shift-invariant (time-invariant) systems, the Mellin transform is the corresponding tool for scale-invariant systems: the logarithmic mapping it performs turns multiplication, that is scaling, into addition.<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S1051200424005049)</sup><sup> • </sup><sup>[11](https://www.sciencedirect.com/topics/engineering/mellin-transforms)</sup> Mellin convolution, \( y(\tau) = \int_{0}^{\infty} x(\tau\eta)\,g(\eta)\,d\eta/\eta \), expresses the input-output relation of scale-invariant systems.<sup>[12](https://doi.org/10.1016/j.dsp.2024.104490)</sup>

Documented applications span several fields:<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S1051200424005049)</sup>

- analysis of algorithms, where harmonic sums and divide-and-conquer recurrences are the natural objects<sup>[3](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)</sup>;
- image processing, including image encryption, watermarking, forgery detection, and object detection;
- radar theory, for computing Cramér-Rao bounds for velocity estimation;
- mathematical finance, where Mellin techniques are used for European and American option pricing and option valuation under jump-diffusion models;
- remote sensing, geophysical systems, traffic modeling, and fractional calculus.

Combining the Mellin transform with the Fourier transform gives the Fourier–Mellin transform, which provides simultaneous invariance to scale and position; in image analysis it underlies registration and recognition methods invariant to translation, rotation, and scaling.<sup>[11](https://www.sciencedirect.com/topics/engineering/mellin-transforms)</sup><sup> • </sup><sup>[13](https://www.sciencedirect.com/science/article/pii/S1077314201909221)</sup> In pattern recognition, the magnitude of Fourier–Mellin features is invariant to rotation and scale change, which has made it a standard component of Euclidean-invariant shape recognition systems.<sup>[14](https://arxiv.org/abs/2203.06787)</sup>

## Related transforms

The Mellin transform sits in a family of classical integral transforms alongside the Laplace, Fourier, and wavelet transforms, each matched to a different invariance: Laplace methods for transient and initial-value problems, Fourier methods for oscillatory and spectral phenomena, wavelet methods for localized multiscale behavior, and Mellin methods for scale-sensitive analysis.<sup>[15](https://iaeme.com/Home/article_id/IJARET_09_05_022)</sup> It can be viewed as the [Laplace transform](https://www.edgepedia.org/laplace-transform) evaluated on the imaginary axis after a logarithmic change of variable, and a discretized form permits fast numerical computation of expressions involving dilated functions, such as wavelet coefficients.<sup>[16](https://dsp-book.narod.ru/TAH/ch11.pdf)</sup> Numerical inversion of the transform is an active topic in its own right, with methods based on Laguerre-polynomial expansions that convert the Mellin inversion into a Laplace inversion followed by regularization.<sup>[17](https://www.sciencedirect.com/science/article/pii/001046559500074P)</sup>

## References

1. [DLMF §2.5: Mellin Transform Methods, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/2.5)
2. [Unified framework for linear scale invariant signals, systems, and transforms: A tutorial, Digital Signal Processing 157 (2025), 104880](https://www.sciencedirect.com/science/article/abs/pii/S1051200424005049)
3. [P. Flajolet, P. V. Flajolet et al., Section 2.4 Mellin Transforms, in Average-Case Analysis of Algorithms and Data Structures](https://algo.inria.fr/flajolet/Publications/ViFl90.pdf)
4. [Noam Elkies, Riemann's zeta function and the functional equation, Harvard M259 notes](https://people.math.harvard.edu/~elkies/M259.06/zeta1.pdf)
5. [PMATH 944: Modular Forms, lecture notes, University of Waterloo](https://uwaterloo.ca/pure-mathematics/sites/default/files/uploads/documents/pmath_944_lecture_notes_-_modular_forms.pdf)
6. [Asymptotic Expansion Using Mellin Transform, UC eScholarship](https://escholarship.org/content/qt8z61b9g5/qt8z61b9g5_noSplash_f53bec10912b7dddf2fb5abc06c09064.pdf?t=rcwcwe)
7. [Mellin Transforms and Scaling Asymptotics, QFT.org](https://qft.org/mathematical-methods/asymptotic-special-functions/mellin-transforms-scaling-asymptotics/)
8. [Riemann's Zeta Function and Beyond, Rutgers (Gelbart–Miller)](https://sites.math.rutgers.edu/~sdmiller/gelbart-miller-zeta.pdf)
9. [Modular Forms, course notes, University of Oklahoma](https://math.ou.edu/~kmartin/mfs/mfs.pdf)
10. [Mellin publications, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Mellin_publications/)
11. [Mellin Transforms: an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/engineering/mellin-transforms)
12. [Principles of fractional signal processing, Digital Signal Processing (2024)](https://doi.org/10.1016/j.dsp.2024.104490)
13. [Robust and Efficient Fourier–Mellin Transform Approximations for Gray-Level Image Reconstruction, Journal of Visual Communication and Image Representation](https://www.sciencedirect.com/science/article/pii/S1077314201909221)
14. [Euclidean Invariant Recognition of 2D Shapes Using Histograms of Magnitudes of Local Fourier-Mellin Descriptors, arXiv:2203.06787](https://arxiv.org/abs/2203.06787)
15. [Applications of Integral Transforms in Engineering and Applied Sciences: A Narrative Review, IJARET 9(5), 2018](https://iaeme.com/Home/article_id/IJARET_09_05_022)
16. [Chapter 11: The Mellin Transform, in Transform Analysis of Signals](https://dsp-book.narod.ru/TAH/ch11.pdf)
17. [On regularized numerical inversion of Mellin transform, Journal of Computational Physics](https://www.sciencedirect.com/science/article/pii/001046559500074P)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Analytic number theory › Zeta and L-functions › Modular forms and L-function interface*

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