Integration by Mellin transforms¶

Method -> "Mellin" evaluates half-line definite integrals

\[ M_f(s) \;=\; \int_0^{\infty} x^{\,s-1}\, f(x)\, dx \]

in closed form. The quantity \(M_f(s)\) is the Mellin transform of \(f\), and a great many named constants and special functions are, at heart, Mellin transforms evaluated at a particular \(s\) — the Gamma function, the Beta function, \(\pi/2\) (Dirichlet's integral), and the whole hypergeometric family among them.

This method is the engine behind Mathilda's IntegrateRamanujanMasterTheorembuiltin;"Mellin"and"RamanujanMasterTheorem"` are two names for the same method.

In[1]:= Integrate[x^(s - 1) Exp[-x], {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= ConditionalExpression[Gamma[s], s > 0]

That one line is the definition of the Gamma function, and Mathilda returns it together with the exact strip of convergence, s > 0, wrapped in a ConditionalExpression.

1. The mathematical idea¶

Mellin transforms¶

The Mellin transform \(M_f(s)=\int_0^\infty x^{s-1}f(x)\,dx\) is the multiplicative analogue of the Laplace/Fourier transform (substitute \(x=e^{-t}\) and it becomes a two-sided Laplace transform). It converges on a vertical strip \(\alpha < \operatorname{Re} s < \beta\) in the complex \(s\)-plane, whose edges are fixed by the growth of \(f\) at the two ends of the ray:

if \(f(x)\sim x^{-\alpha}\) as \(x\to 0^+\), the integral needs \(\operatorname{Re} s > \alpha\) for convergence at the origin;
if \(f(x)\sim x^{-\beta}\) as \(x\to\infty\), it needs \(\operatorname{Re} s < \beta\) for convergence at infinity.

A closed form is only valid inside this strip, which is exactly why Mathilda reports it. Outside the strip the same algebraic expression is the analytic continuation of the integral, not the integral itself.

The Ramanujan Master Theorem¶

For an integrand given by a power series \(f(x)=\sum_{k\ge 0}\frac{(-1)^k}{k!}\,\varphi(k)\,x^{k}\), Ramanujan's Master Theorem (RMT) states

\[ \int_0^\infty x^{\,s-1} f(x)\,dx \;=\; \Gamma(s)\,\varphi(-s), \]

turning integration into analytic interpolation of the coefficients: read off \(\varphi(k)\) from the series, replace \(k \mapsto -s\), and multiply by \(\Gamma(s)\). For a generalized hypergeometric integrand \({}_pF_q(\mathbf a;\mathbf b;-\lambda x)\) the coefficient function is \(\varphi(k)=\prod_i (a_i)_k \big/ \prod_j (b_j)_k\), and RMT delivers the ratio-of-Gammas transform in one step — see §4.

References¶

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford, 1948 — Mellin inversion and convergence strips (Ch. VII).
G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge, 1940 — Ch. XI, the Master Theorem.
T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Harrison, V. H. Moll, A. Straub, "Ramanujan's Master Theorem," The Ramanujan Journal 29 (2012) 103–120.
P. Flajolet, X. Gourdon, P. Dumas, "Mellin transforms and asymptotics: Harmonic sums," Theoretical Computer Science 144 (1995) 3–58.
F. Oberhettinger, Tables of Mellin Transforms, Springer, 1974.
NIST Digital Library of Mathematical Functions, §1.14 (integral transforms), dlmf.nist.gov/1.14.

2. How Mathilda realises it¶

The method is a table plus a small set of operational rules, applied symbolically. There is no numerical quadrature anywhere in it — every answer is correct by construction.

Split into terms. The integrand is expanded into a sum \(\sum_i c_i\, x^{\rho_i}\, \mathrm{kernel}_i(x)\). Each additive term is integrated independently; because every surviving term converges absolutely on its own strip, the sum of the integrals equals the integral of the sum.
Peel off the power prefactor. Each term is written as \(C\,x^{\rho}\,\mathrm{kernel}(x)\) with \(C\) free of \(x\). The pure power is absorbed into the transform variable by setting \(s = \rho + 1\): the leading \(x^{s-1}\) is just bookkeeping for "which point of \(M_{\mathrm{kernel}}\) do we want".
Match a base kernel. The single remaining factor is matched against a table of proven base Mellin transforms (see §7). A linear internal scaling \(x \mapsto \lambda x\) folds into the transform as the \(\lambda^{-s}\) factor; a monomial substitution \(x \mapsto x^{k}\) (as in Cos[Sqrt[x]]) rescales the strip and the argument of \(\Gamma\) accordingly.
Gate on the convergence strip. Every table identity carries its strip. Mathilda reduces the strip against your Assumptions (or evaluates it numerically when \(s\) is a number). If the strip is proven, the bare closed form is returned; otherwise it is carried as a ConditionalExpression. This gate is what makes the result unconditionally correct rather than a formal manipulation.

A product of two transcendental kernels (say \(e^{-x}\cos x\)) is a Mellin convolution — Meijer-\(G\) territory — and is deliberately out of scope; the method returns the integral unevaluated rather than guess (see §8).

3. Invoking the method¶

Pin the method with Method -> "Mellin". The integrand must be presented in the half-line Mellin shape x^(s-1) * kernel, over {x, 0, Infinity}.

You do not have to keep s symbolic. If the prefactor is a concrete power, the strip is checked numerically and a bare value comes back:

In[1]:= Integrate[Exp[-x], {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= 1

In[2]:= Integrate[x^2 Exp[-x], {x, 0, Infinity}, Method -> "Mellin"]
Out[2]= 2

Here x^2 Exp[-x] has \(s = 3\), so the answer is \(\Gamma(3) = 2! = 2\).

When s is symbolic, supply the strip yourself through Assumptions and Mathilda will discharge the ConditionalExpression for you:

In[1]:= Integrate[x^(s - 1) Exp[-x], {x, 0, Infinity},
          Method -> "Mellin", Assumptions -> s > 0]
Out[1]= Gamma[s]

Without the assumption the same call returns ConditionalExpression[Gamma[s], s > 0] — the closed form annotated with the condition under which it holds.

4. The Ramanujan Master Theorem (hypergeometric integrands)¶

The RMT layer handles integrands built from a generalized hypergeometric function \({}_pF_q\) with a linear argument, provided \(p \le q + 1\). For \({}_1F_1\) (Kummer's confluent function) the transform is a clean ratio of Gammas:

In[1]:= Integrate[x^(s - 1) Hypergeometric1F1[a, c, -x], {x, 0, Infinity},
          Method -> "Mellin"]
Out[1]= ConditionalExpression[(Gamma[c] Gamma[s] Gamma[a - s])/(Gamma[a] Gamma[c - s]),
          s > 0 && s < a]

and for the Gauss function \({}_2F_1\) there is simply one more Gamma factor top and bottom:

In[1]:= Integrate[x^(s - 1) Hypergeometric2F1[a, b, c, -x], {x, 0, Infinity},
          Method -> "Mellin"]
Out[1]= ConditionalExpression[
          (Gamma[c] Gamma[s] Gamma[a - s] Gamma[b - s])/(Gamma[a] Gamma[b] Gamma[c - s]),
          s > 0 && s < a && s < b]

These are the general pattern

\[ M_{{}_pF_q}(s)=\frac{\prod_j\Gamma(b_j)}{\prod_i\Gamma(a_i)}\;\Gamma(s)\; \frac{\prod_i\Gamma(a_i-s)}{\prod_j\Gamma(b_j-s)}\;\lambda^{-s}, \]

valid on \(0 < \operatorname{Re} s < \min_i \operatorname{Re} a_i\), produced directly by \(\varphi(-s)\).

Many special functions are hypergeometric in disguise. Mathilda rewrites a few of them (Erf, Erfc, the lower incomplete Gamma, spherical Bessel) into \({}_pF_q\) form first, so their Mellin transforms come out of the same machine:

In[1]:= Integrate[x^(s - 1) Erf[x], {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= ConditionalExpression[-Gamma[1/2 (1 + s)]/(Sqrt[Pi] s),
          1/2 (1 + s) > 0 && 1/2 (1 + s) < 1/2]

5. Worked examples¶

Every transcript below is reproduced from the current build. Timings (from // Timing) are indicative only.

5.1 The classical transforms¶

The Gamma and Beta functions are the two workhorse Mellin transforms. The Beta integral is the transform of an algebraic binomial \((1+x)^{-a}\):

In[1]:= Integrate[x^(s - 1)/(1 + x)^a, {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= ConditionalExpression[Beta[s, a - s], s > 0 && s < a]

In[2]:= Integrate[x^(s - 1)/(1 + x), {x, 0, Infinity}, Method -> "Mellin"]
Out[2]= ConditionalExpression[Beta[s, 1 - s], s > 0 && s < 1]

The second is the \(a=1\) special case; recall \(B(s,1-s)=\pi/\sin(\pi s)\) — the reflection formula in integral form.

The Gaussian is handled by a monomial substitution \(x \mapsto x^2\), which halves the argument of \(\Gamma\) and the strip:

In[1]:= Integrate[Exp[-x^2], {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= 1/2 Sqrt[Pi]

In[2]:= Integrate[x^(s - 1) Exp[-x^2], {x, 0, Infinity}, Method -> "Mellin"]
Out[2]= ConditionalExpression[1/2 Gamma[1/2 s], s > 0]

5.2 Trigonometric kernels¶

The Mellin transforms of \(\cos\) and \(\sin\) are \(\Gamma(s)\cos(\pi s/2)\) and \(\Gamma(s)\sin(\pi s/2)\) on their strips. Mathilda returns them in the equivalent reflection form (a \(\pi\csc/\Gamma\) or \(\pi\sec/\Gamma\) ratio), which is the numerically robust representation near the strip edges:

In[1]:= Integrate[x^(s - 1) Cos[x], {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= ConditionalExpression[(1/2 Pi Csc[1/2 Pi s])/Gamma[1 - s], s > 0 && s < 1]

The Dirichlet integral \(\int_0^\infty \frac{\sin x}{x}\,dx=\frac{\pi}{2}\) falls straight out as the \(s=1\) slice of the \(\sin x / x\) transform:

In[1]:= Integrate[x^(s - 1) Sin[x]/x, {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= ConditionalExpression[(1/2 Pi Sec[1/2 Pi (-1 + s)])/Gamma[2 - s], s > 0 && s < 2]

In[2]:= Integrate[Sin[x]/x, {x, 0, Infinity}, Method -> "Mellin"]
Out[2]= 1/2 Pi

Sin[x]/x at \(s=1\) hits a removable \(0\cdot\infty\) coincidence in the raw formula, which Mathilda resolves with a limit before returning \(\pi/2\).

5.3 The showcase set¶

The following integrals exercise every branch of the method — internal monomial substitutions, the logarithm and arctangent kernels, Bessel functions, and the hypergeometric layer. Each is returned with its exact strip.

Exponential — the Gamma function.

In[1]:= Integrate[x^(s - 1) Exp[-x], {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.001454, ConditionalExpression[Gamma[s], s > 0]}

Algebraic binomial — the Beta function.

In[1]:= Integrate[x^(s - 1)/(1 + x)^a, {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.001405, ConditionalExpression[Beta[s, a - s], s > 0 && s < a]}

A logarithm kernel. Log[1 + x]/x has \(\rho=-1\), shifting \(s\to s-1\) inside the \(\pi\csc\) transform:

In[1]:= Integrate[x^(s - 1) Log[1 + x]/x, {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.147829, ConditionalExpression[(Pi Csc[Pi (-1 + s)])/(-1 + s), s > 0 && s < 1]}

Trigonometric kernel with a square-root argument. Cos[Sqrt[x]] triggers the \(x \mapsto x^2\) monomial substitution, doubling the strip's slope (\(2s < 1\)) and the \(\Gamma\) argument:

In[1]:= Integrate[x^(s - 1) Cos[Sqrt[x]], {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.057947, ConditionalExpression[(Pi Csc[Pi s])/Gamma[1 - 2 s], s > 0 && 2 s < 1]}

Another square-root argument, with a prefactor.

In[1]:= Integrate[x^(s - 1) Sin[Sqrt[x]]/Sqrt[x], {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.246038, ConditionalExpression[(Pi Sec[-1/2 Pi + Pi s])/Gamma[2 - 2 s], s > 0 && s < 1]}

A Bessel-function kernel. The Weber–Schafheitlin-type transform of \(J_\nu(2\sqrt{x})\,x^{-\nu/2}\) collapses to a ratio of two Gammas:

In[1]:= Integrate[x^(s - 1) BesselJ[nu, 2 Sqrt[x]]/x^(nu/2), {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.012294, ConditionalExpression[Gamma[s]/Gamma[1 + nu - s], s > 0 && -3/2 + 2 s < nu]}

An arctangent kernel.

In[1]:= Integrate[x^(s - 1) ArcTan[Sqrt[x]]/Sqrt[x], {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.31456, ConditionalExpression[-(Pi Sec[-1/2 Pi + Pi s])/(-1 + 2 s), s > 0 && 2 s < 1]}

Confluent hypergeometric — Ramanujan's Master Theorem.

In[1]:= Integrate[x^(s - 1) Hypergeometric1F1[a, c, -x], {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.008704, ConditionalExpression[(Gamma[c] Gamma[s] Gamma[a - s])/(Gamma[a] Gamma[c - s]),
          s > 0 && s < a]}

Gauss hypergeometric — Ramanujan's Master Theorem.

In[1]:= Integrate[x^(s - 1) Hypergeometric2F1[a, b, c, -x], {x, 0, Infinity}, Method -> "Mellin"] // Timing
Out[1]= {0.01787, ConditionalExpression[
          (Gamma[c] Gamma[s] Gamma[a - s] Gamma[b - s])/(Gamma[a] Gamma[b] Gamma[c - s]),
          s > 0 && s < a && s < b]}

6. Convergence and `ConditionalExpression`¶

When s (or any other parameter) is symbolic, the closed form is only valid inside the convergence strip, so Mathilda returns a ConditionalExpression[value, condition]. The condition is the strip, reduced against your Assumptions:

Supply Assumptions -> (0 < s < a) and the whole strip is discharged — the bare value is returned.
Supply a partial assumption and the residual, still-undischarged part of the strip is carried in the ConditionalExpression.
Supply nothing and you get the full strip back, as above.

A ConditionalExpression collapses to its value automatically once the condition is known to hold (for instance when the parameters later become numbers that satisfy it). Treat the condition as an honest statement of where the formula is the integral — outside the strip it is the analytic continuation, not the integral itself.

7. The kernel table¶

The base transforms the method knows, each with its convergence strip (\(s\) = Mellin variable, \(\lambda,c > 0\)):

Kernel \(f(x)\)	Mellin transform \(M_f(s)\)	Strip
\(e^{c x}\)	\(\Gamma(s)\,(-c)^{-s}\)	\(0 < \operatorname{Re} s\)
\(e^{c x^2}\)	\(\tfrac12(-c)^{-s/2}\,\Gamma(s/2)\)	\(0 < \operatorname{Re} s\)
\((p+q x^{m})^{-a}\)	\(\tfrac1m\, p^{s/m-a}\, q^{-s/m}\,B(s/m,\,a-s/m)\)	\(0 < \operatorname{Re} s < m\operatorname{Re} a\)
\(\cos(\lambda x)\)	\(\Gamma(s)\cos(\pi s/2)\,\lambda^{-s}\)	\(0 < \operatorname{Re} s < 1\)
\(\sin(\lambda x)\)	\(\Gamma(s)\sin(\pi s/2)\,\lambda^{-s}\)	\(-1 < \operatorname{Re} s < 1\)
\(\log(1+\lambda x)\)	\(\dfrac{\pi}{s\,\sin(\pi s)}\,\lambda^{-s}\)	\(-1 < \operatorname{Re} s < 0\)
\(\arctan(\lambda x)\)	\(-\dfrac{\pi}{2 s\,\cos(\pi s/2)}\,\lambda^{-s}\)	\(-1 < \operatorname{Re} s < 0\)
\(J_\nu(\lambda x)\)	\(2^{s-1}\lambda^{-s}\,\dfrac{\Gamma((\nu+s)/2)}{\Gamma((\nu-s)/2+1)}\)	\(-\operatorname{Re}\nu < \operatorname{Re} s < 3/2\)
\(\operatorname{Li}_\nu(-\lambda x)\)	\(-\pi(-s)^{-\nu}\lambda^{-s}/\sin(\pi s)\)	\(-1 < \operatorname{Re} s < 0\)
\({}_pF_q(\mathbf a;\mathbf b;-\lambda x)\), \(p\le q{+}1\)	\(\dfrac{\prod_j\Gamma(b_j)}{\prod_i\Gamma(a_i)}\,\Gamma(s)\,\dfrac{\prod_i\Gamma(a_i-s)}{\prod_j\Gamma(b_j-s)}\,\lambda^{-s}\)	\(0<\operatorname{Re} s<\min_i\operatorname{Re} a_i\)

Two operational rules extend the table:

Power prefactor \(x^\rho\): shifts the transform argument, \(s \mapsto s+\rho\) (equivalently, sets \(s=\rho+1\) for a bare kernel).
Monomial substitution \(x \mapsto x^{k}\): replaces \(s\) by \(s/k\) and rescales the strip — this is how Cos[Sqrt[x]], Sin[Sqrt[x]]/Sqrt[x], and Exp[-x^2] are reduced to table entries.

8. Limitations¶

One transcendental kernel per term. A product of two transcendental factors is a Mellin convolution and is out of scope. Such an integral is returned unevaluated rather than approximated:

In[1]:= Integrate[x^(s - 1) Exp[-x] Cos[x], {x, 0, Infinity}, Method -> "Mellin"]
Out[1]= Integrate[Cos[x] E^(-x) x^(-1 + s), {x, 0, Infinity}, Method -> "Mellin"]

(This particular integral does have a closed form via a contour or Laplace-transform argument; the Mellin table method simply does not carry convolution rules.)

Half-line only. The method integrates over {x, 0, Infinity}. For finite or doubly-infinite ranges use the other methods in the cascade ("NewtonLeibniz", "Residue", "LineIntegral", …).
The strip must be provable. If neither the numeric value of s nor your Assumptions can settle the convergence strip, the strip is carried as a ConditionalExpression; the method never silently drops a convergence condition.