--- title: "MellinTransform" language: "en" type: "Symbol" summary: "MellinTransform[expr, x, s] gives the Mellin transform of expr. MellinTransform[expr, {x1, x2, ...}, {s1, s2, ...}] gives the multidimensional Mellin transform of expr." keywords: - function transform - integral operator - integral transform - Integrate - inverse Mellin transform - Mellin transform - Mellin convolution - Meijer G function - inttrans - mellin canonical_url: "https://reference.wolfram.com/language/ref/MellinTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" - title: "Summation Transforms" link: "https://reference.wolfram.com/language/guide/SummationTransforms.en.md" related_functions: - title: "InverseMellinTransform" link: "https://reference.wolfram.com/language/ref/InverseMellinTransform.en.md" - title: "MellinConvolve" link: "https://reference.wolfram.com/language/ref/MellinConvolve.en.md" - title: "LaplaceTransform" link: "https://reference.wolfram.com/language/ref/LaplaceTransform.en.md" - title: "BilateralLaplaceTransform" link: "https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.en.md" - title: "FourierTransform" link: "https://reference.wolfram.com/language/ref/FourierTransform.en.md" - title: "HankelTransform" link: "https://reference.wolfram.com/language/ref/HankelTransform.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" - title: "MeijerG" link: "https://reference.wolfram.com/language/ref/MeijerG.en.md" - title: "FoxH" link: "https://reference.wolfram.com/language/ref/FoxH.en.md" - title: "Gamma" link: "https://reference.wolfram.com/language/ref/Gamma.en.md" - title: "MeijerGReduce" link: "https://reference.wolfram.com/language/ref/MeijerGReduce.en.md" - title: "FoxHReduce" link: "https://reference.wolfram.com/language/ref/FoxHReduce.en.md" - title: "Asymptotic" link: "https://reference.wolfram.com/language/ref/Asymptotic.en.md" --- # MellinTransform MellinTransform[expr, x, s] gives the Mellin transform of expr. MellinTransform[expr, {x1, x2, …}, {s1, s2, …}] gives the multidimensional Mellin transform of expr. ## Details and Options * The Mellin transform of a function $f(x)$ is defined to be $\int_0^{\infty } f(x) x^{s-1} \, dx$. * The multidimensional Mellin transform of a function $f\left(x_1,x_2,\ldots \right)$ is given by $\int _0^{\infty }\int _0^{\infty }\cdots f\left(x_1,x_2,\ldots \right)x_1^{s_1-1} x_2^{s_2-1}\ldots dx_1dx_2\ldots$. * The Mellin transform of $f(x)$ exists only for complex values of $s$ such that $\alpha <\Re(s)<\beta$. In some cases, this strip of definition may extend to a half-plane. [image] * The following options can be given: | | | | | ------------------- | ------------- | ----------------------------------------------------------------- | | Assumptions | \$Assumptions | assumptions on parameters | | GenerateConditions | False | whether to generate results that involve conditions on parameters | | Method | Automatic | what method to use | * In ``TraditionalForm``, ``MellinTransform`` is output using $\mathcal{M}_x[f(x)](s)$. --- ## Examples (39) ### Basic Examples (2) Compute the Mellin transform of a function: ```wl In[1]:= MellinTransform[E ^ (-a x), x, s] Out[1]= a^-s Gamma[s] ``` --- Compute a multivariate Mellin transform: ```wl In[1]:= MellinTransform[Sin[x - y ^ 2], {x, y}, {s, t}] Out[1]= (1/2) Cos[(1/4) π (2 - 2 s + t)] Gamma[s] Gamma[(t/2)] ``` ### Scope (16) #### Basic Uses (3) Compute the Mellin transform of a function for a symbolic parameter ``s`` : ```wl In[1]:= MellinTransform[1 / (1 + x), x, s] Out[1]= π Csc[π s] ``` Use an exact value for the parameter: ```wl In[2]:= MellinTransform[1 / (1 + x), x, 1 / 2] Out[2]= π ``` Use an inexact value for the parameter: ```wl In[3]:= MellinTransform[1 / (1 + x), x, 0.3] Out[3]= 3.88322 ``` --- Obtain the condition for validity of a Mellin transform: ```wl In[1]:= MellinTransform[E ^ (-x), x, s, GenerateConditions -> True] Out[1]= ConditionalExpression[Gamma[s], Re[s] > 0] ``` The result is valid in the half-plane $\Re(s)>0$ : [image] --- ``TraditionalForm`` formatting: ```wl In[1]:= MellinTransform[f[x], x, s]//TraditionalForm Out[1]//TraditionalForm= $$\mathcal{M}_x[f(x)](s)$$ ``` #### Elementary Functions (3) Exponential function: ```wl In[1]:= MellinTransform[E ^ (-a x), x, s] Out[1]= a^-s Gamma[s] ``` Gaussian function: ```wl In[2]:= MellinTransform[E ^ (-x ^ 2), x, s] Out[2]= (1/2) Gamma[(s/2)] ``` General exponential functions: ```wl In[3]:= MellinTransform[Exp[-x - x ^ (-1)], x, s] Out[3]= 2 BesselK[-s, 2] In[4]:= MellinTransform[1 / (E ^ x - 1), x, s] Out[4]= Gamma[s] Zeta[s] ``` Composition of logarithmic and exponential functions: ```wl In[5]:= MellinTransform[Log[1 - E ^ (-x)], x, s] Out[5]= -Gamma[s] PolyLog[1 + s, 1] ``` --- Rational functions: ```wl In[1]:= MellinTransform[(1/3x + 5), x, s] Out[1]= 3^-s 5^-1 + s π Csc[π s] In[2]:= MellinTransform[(1/x ^ 2 + 1), x, s] Out[2]= (1/2) π Csc[(π s/2)] ``` --- Mellin transforms of polynomials are given in terms of ``DiracDelta`` : ```wl In[1]:= MellinTransform[x, x, s] Out[1]= 2 π DiracDelta[I (1 + s)] In[2]:= MellinTransform[x ^ 2 + 3x, x, s] Out[2]= 6 π DiracDelta[I (1 + s)] + 2 π DiracDelta[I (2 + s)] ``` #### Special Functions (3) ``MellinTransform`` of ``BesselJ`` : ```wl In[1]:= MellinTransform[BesselJ[1, x], x, s] Out[1]= (2^-1 + s Gamma[(1/2) + (s/2)]/Gamma[(3/2) - (s/2)]) ``` ``BesselK`` : ```wl In[2]:= MellinTransform[BesselK[0, x], x, s] Out[2]= 2^-2 + s Gamma[(s/2)]^2 ``` Product of Bessel functions: ```wl In[3]:= MellinTransform[BesselJ[0, x]BesselJ[1, x], x, s] Out[3]= (Gamma[1 - (s/2)] Gamma[(1/2) + (s/2)]/2 Sqrt[π] Gamma[(3/2) - (s/2)]^2) ``` --- Exponential integral function ``ExpIntegralE`` : ```wl In[1]:= MellinTransform[ExpIntegralE[n, x], x, s] Out[1]= (Gamma[s]/-1 + n + s) ``` ``ExpIntegralEi`` : ```wl In[2]:= MellinTransform[ExpIntegralEi[-x], x, s] Out[2]= -(Gamma[s]/s) ``` --- ``MellinTransform`` of the error function ``Erf`` : ```wl In[1]:= MellinTransform[Erf[x], x, s] Out[1]= -(Gamma[(1/2) + (s/2)]/Sqrt[π] s) ``` Complementary error function ``Erfc`` : ```wl In[2]:= MellinTransform[Erfc[x], x, s] Out[2]= (Gamma[(1/2) + (s/2)]/Sqrt[π] s) ``` #### Piecewise Functions (3) ``MellinTransform`` of ``UnitStep`` : ```wl In[1]:= MellinTransform[UnitStep[x - 1], x, s] Out[1]= -(1/s) ``` ``UnitBox`` : ```wl In[2]:= MellinTransform[UnitBox[x - 1], x, s] Out[2]= (2^-s (-1 + 3^s)/s) ``` ``UnitTriangle`` : ```wl In[3]:= MellinTransform[UnitTriangle[x - 3], x, s] Out[3]= (2^1 + s - 2 3^1 + s + 4^1 + s/s + s^2) ``` --- Products of functions with ``UnitStep`` : ```wl In[1]:= MellinTransform[x Log[x]UnitStep[1 - x], x, s] Out[1]= -(1/(1 + s)^2) In[2]:= MellinTransform[Sin[Log[x]]UnitStep[x - 1], x, s] Out[2]= (1/1 + s^2) ``` --- ``MellinTransform`` of a ``Piecewise`` function: ```wl In[1]:= MellinTransform[Piecewise[{{E ^ (-x ^ 2), 1 < x < 2}}], x, s] Out[1]= (1/2) (Gamma[(s/2), 1] - Gamma[(s/2), 4]) ``` #### Generalized Functions (2) ``MellinTransform`` of functions involving ``HeavisideTheta`` : ```wl In[1]:= MellinTransform[HeavisideTheta[x - 1], x, s] Out[1]= -(1/s) In[2]:= MellinTransform[x HeavisideTheta[x - 1], x, s] Out[2]= -(1/1 + s) ``` --- ``DiracDelta`` : ```wl In[1]:= MellinTransform[DiracDelta[x - 1], x, s] Out[1]= 1 ``` #### Multivariate Functions (2) Multivariate rational function: ```wl In[1]:= MellinTransform[1 / (x + y ^ 2 + 1), {x, y}, {s, t}] Out[1]= (π Csc[(π t/2)] Gamma[s] Gamma[1 - s - (t/2)]/2 Gamma[1 - (t/2)]) ``` --- Multivariate exponential function: ```wl In[1]:= MellinTransform[E ^ (-x - 3y ^ 2), {x, y}, {s, t}] Out[1]= (1/2) 3^-t / 2 Gamma[s] Gamma[(t/2)] ``` ### Options (5) #### Assumptions (1) Compute the Mellin transform of a function depending on a parameter ``a`` : ```wl In[1]:= MellinTransform[E ^ (-x)UnitStep[x - a], x, s] Out[1]= Piecewise[{{Gamma[s], a <= 0}}, Gamma[s, a]] ``` Obtain a simpler result by specifying assumptions on the parameter: ```wl In[2]:= MellinTransform[E ^ (-x)UnitStep[x - a], x, s, Assumptions -> a > 0] Out[2]= Gamma[s, a] ``` #### GenerateConditions (1) Obtain conditions for validity of the result given by ``MellinTransform`` : ```wl In[1]:= MellinTransform[E ^ (-x), x, s, GenerateConditions -> True] Out[1]= ConditionalExpression[Gamma[s], Re[s] > 0] ``` ``GenerateConditions`` is set to ``False`` by default in this case: ```wl In[2]:= MellinTransform[E ^ (-x), x, s] Out[2]= Gamma[s] ``` #### Method (3) Compute a Mellin transform using the default method: ```wl In[1]:= MellinTransform[1 / (E ^ x - 1), x, s] Out[1]= Gamma[s] Zeta[s] ``` This example is done using table lookup: ```wl In[2]:= MellinTransform[1 / (E ^ x - 1), x, s, Method -> "TableLookUp"] Out[2]= Gamma[s] Zeta[s] ``` Attempting to evaluate this example by a conversion to ``MeijerG`` fails: ```wl In[3]:= MellinTransform[1 / (E ^ x - 1), x, s, Method -> "MeijerG"] Out[3]= MellinTransform[(1/-1 + E^x), x, s, Method -> "MeijerG"] ``` Evaluate the example using the definition of ``MellinTransform`` in terms of ``Integrate`` : ```wl In[4]:= MellinTransform[1 / (E ^ x - 1), x, s, Method -> "Integrate"] Out[4]= Gamma[s] PolyLog[s, 1] ``` --- The default method uses a conversion to ``MeijerG`` for this example: ```wl In[1]:= MellinTransform[BesselJ[1, x] ^ 2, x, s]//Timing Out[1]= {0.0156001, (Gamma[(1/2) - (s/2)] Gamma[1 + (s/2)]/2 Sqrt[π] Gamma[1 - (s/2)] Gamma[2 - (s/2)])} In[2]:= MellinTransform[BesselJ[1, x] ^ 2, x, s, Method -> "MeijerG"]//Timing Out[2]= {0., (Gamma[(1/2) - (s/2)] Gamma[1 + (s/2)]/2 Sqrt[π] Gamma[1 - (s/2)] Gamma[2 - (s/2)])} ``` This is faster than using the definition of ``MellinTransform`` in terms of ``Integrate`` : ```wl In[3]:= MellinTransform[BesselJ[1, x] ^ 2, x, s, Method -> "Integrate"]//Timing Out[3]= {6.48964, (Gamma[(1/2) - (s/2)] Gamma[1 + (s/2)]/2 Sqrt[π] Gamma[1 - (s/2)] Gamma[2 - (s/2)])} ``` --- Here, the symbolic method fails because the input is purely numerical: ```wl In[1]:= f[x_ ? NumericQ] := E ^ (-x) In[2]:= MellinTransform[f[x], x, 0.3, Method -> "Integrate"] Out[2]= MellinTransform[f[x], x, 0.3, Method -> "Integrate"] ``` This example is evaluated using a numerical method based on ``NIntegrate`` : ```wl In[3]:= MellinTransform[f[x], x, 0.3, Method -> "Numeric"] Out[3]= 2.99157 In[4]:= MellinTransform[f[x], x, 0.3] Out[4]= 2.99157 ``` ### Applications (3) Use ``MellinTransform`` to evaluate $\int_0^{\infty } \frac{J_0(t) J_1\left(\frac{x}{t}\right)}{t} \, dt$, which may be regarded as a Mellin convolution of the following functions: ```wl In[1]:= f[t_] := BesselJ[0, t] In[2]:= g[t_] := BesselJ[1, t] ``` Apply ``MellinTransform`` to each function: ```wl In[3]:= f1 = MellinTransform[f[t] , t, s] Out[3]= (2^-1 + s Gamma[(s/2)]/Gamma[1 - (s/2)]) In[4]:= g1 = MellinTransform[g[t] , t, s] Out[4]= (2^-1 + s Gamma[(1/2) + (s/2)]/Gamma[(3/2) - (s/2)]) ``` Obtain the required integral by performing an inverse Mellin transform: ```wl In[5]:= InverseMellinTransform[f1 g1, s, x] Out[5]= (BesselJ[1, 2 Sqrt[x]]/Sqrt[x]) ``` Compute the integral directly using ``Integrate`` : ```wl In[6]:= Integrate[f[t] g[x / t] / t, {t, 0, Infinity}, Assumptions -> x > 0]//FunctionExpand Out[6]= (BesselJ[1, 2 Sqrt[x]]/Sqrt[x]) ``` Obtain the same result using ``MellinConvolve`` : ```wl In[7]:= MellinConvolve[f[t], g[t], t, x]//FunctionExpand Out[7]= (BesselJ[1, 2 Sqrt[x]]/Sqrt[x]) In[8]:= Plot[{f[x], g[x], %}//Evaluate, {x, 0, 10}, Filling -> Axis] Out[8]= [image] ``` --- Find the general solution of the Bessel equation using ``MellinTransform`` : ```wl In[1]:= besseleqn = x ^ 2 y''[x] + x y'[x] + (x ^ 2 - 1) y[x] == 0; ``` Apply ``MellinTransform`` to the equation: ```wl In[2]:= MellinTransform[besseleqn, x, s]//Expand Out[2]= -MellinTransform[y[x], x, s] + s^2 MellinTransform[y[x], x, s] + MellinTransform[y[x], x, 2 + s] == 0 ``` Use ``RSolveValue`` to solve the recurrence equation: ```wl In[3]:= RSolveValue[%, MellinTransform[y[x], x, s], s] Out[3]= -(I^s 2^-1 + s C[1] Gamma[(1/2) (-1 + s)] Gamma[(1 + s/2)]/π) + ((-1)^1 + s I^s 2^-1 + s C[2] Gamma[(1/2) (-1 + s)] Gamma[(1 + s/2)]/π) ``` Use ``InverseMellinTransform`` to find the required general solution: ```wl In[4]:= (Simplify[InverseMellinTransform[%, s, x]//FunctionExpand, x > 0] /. {(C[1] - C[2]) -> -IC[2], (C[1] + C[2]) -> C[1]})//Expand Out[4]= BesselJ[1, x] C[1] + BesselY[1, x] C[2] ``` Verify the result using ``DSolveValue`` : ```wl In[5]:= DSolveValue[x ^ 2 y''[x] + x y'[x] + (x ^ 2 - 1) y[x] == 0, y[x], x] Out[5]= BesselJ[1, x] C[1] + BesselY[1, x] C[2] ``` --- Use ``MellinTransform`` to find the first two terms in the asymptotic expansion for a function that is defined by an infinite series: ```wl In[1]:= f[x_] := Inactive[Sum][((-1)^k*Log[k])/E^(k^2*x), {k, 1, Infinity}] ``` Compute the Mellin transform of $f(x)$ : ```wl In[2]:= MellinTransform[f[x], x, s] Out[2]= Inactive[Sum][((-1)^k*Gamma[s]*Log[k])/(k^2)^s, {k, 1, Infinity}] In[3]:= Activate[%] Out[3]= 4^-s Gamma[s] (Log[4] Zeta[2 s] + (-2 + 4^s) Derivative[1][Zeta][2 s]) ``` Compute the residues at $s=0$ and $s=-1$ to obtain the required asymptotic expansion: ```wl In[4]:= Residue[% x ^ (-s), {s, 0}] + Residue[% x ^ (-s), {s, -1}]//Simplify Out[4]= (1/2) Log[(π/2)] + 7 x Derivative[1][Zeta][-2] ``` ### Properties & Relations (11) Use ``Asymptotic`` to compute an asymptotic approximation: ```wl In[1]:= Asymptotic[Inactive[MellinTransform][E ^ (-x), x, s], s -> Infinity] Out[1]= E^-s Sqrt[2 π] s^-(1/2) + s ``` --- ``MellinTransform`` computes the integral $\int _0^{\infty }e^{-\text{\textit{$x$}}} \text{\textit{$x$}}^{\text{\textit{$s$}}-1}d\text{\textit{$x$}}$ : ```wl In[1]:= MellinTransform[E ^ (-x), x, s] Out[1]= Gamma[s] ``` Obtain the same result using ``Integrate`` : ```wl In[2]:= Integrate[ E ^ (-x) x ^ (s - 1), {x, 0, ∞}, Assumptions -> s > 0] Out[2]= Gamma[s] ``` --- ``MellinTransform`` and ``InverseMellinTransform`` are mutual inverses: ```wl In[1]:= InverseMellinTransform[MellinTransform[f[x], x, s], s, x] Out[1]= f[x] In[2]:= MellinTransform[InverseMellinTransform[g[s], s, x], x, s] Out[2]= g[s] ``` Verify the relationship for a specific function: ```wl In[3]:= MellinTransform[E ^ (-x), x, s] Out[3]= Gamma[s] In[4]:= InverseMellinTransform[%, s, x] Out[4]= E^-x ``` --- ``MellinTransform`` is a linear operator: ```wl In[1]:= MellinTransform[a f[x] + b g[x], x, s] Out[1]= a MellinTransform[f[x], x, s] + b MellinTransform[g[x], x, s] ``` --- The Mellin transform of $x^n f^{(n)}(x)$ is given by $(-1)^n (s)_n \mathcal{M}_x[f(x)](s)$ : ```wl In[1]:= MellinTransform[x f'[x], x, s] Out[1]= -s MellinTransform[f[x], x, s] In[2]:= MellinTransform[x ^ 2 f''[x], x, s] Out[2]= s (1 + s) MellinTransform[f[x], x, s] ``` --- The Mellin transform of $f(a x)$ is given by $\mathcal{M}_x[f(a x)](s)=\frac{\mathcal{M}_x[f(x)](s)}{a^s}$ for positive values of ``a`` : ```wl In[1]:= f[x_] := E ^ (-x) In[2]:= a = 3; In[3]:= MellinTransform[f[a x], x, s] == MellinTransform[f[x], x, s] / a ^ s Out[3]= True ``` --- The Mellin transform of $f\left(x^a\right)$ is given by $\mathcal{M}_x\left[f\left(x^a\right)\right](s)=\frac{\mathcal{M}_x[f(x)]\left(\frac{s}{a}\right)}{| a| }$ for real values of ``a`` : ```wl In[1]:= f[x_] := E ^ (-x) In[2]:= a = 3; In[3]:= MellinTransform[f[x ^ a], x, s] == MellinTransform[f[x], x, s / a] / Abs[a] Out[3]= True ``` --- The Mellin transform of $\int_0^x f(t) \, dt$ is given by $\mathcal{M}_x\left[\int_0^x f(t) \, dt\right](s)=-\frac{\mathcal{M}_x[f(x)](s+1)}{s}$ : ```wl In[1]:= MellinTransform[Integrate[f[t], {t, 0, x}], x, s] Out[1]= -(MellinTransform[f[x], x, 1 + s]/s) ``` --- The Mellin transform of $\int_x^{\infty } f(t) \, dt$ is given by $\mathcal{M}_x\left[\int_x^{\infty } f(t) \, dt\right](s)=\frac{\mathcal{M}_x[f(x)](s+1)}{s}$ : ```wl In[1]:= MellinTransform[Integrate[f[t], {t, x, ∞}], x, s] Out[1]= (MellinTransform[f[x], x, 1 + s]/s) ``` --- The Mellin transform of a Mellin convolution is the product of the individual Mellin transforms: ```wl In[1]:= MellinTransform[MellinConvolve[f[t], g[t], t, x], x, s] Out[1]= MellinTransform[f[x], x, s] MellinTransform[g[x], x, s] ``` Verify the relationship for a specific pair of functions: ```wl In[2]:= f[t_] := 1 / (t + 1) In[3]:= g[t_] := E ^ (-t) In[4]:= MellinTransform[MellinConvolve[f[t], g[t], t, x], x, s] Out[4]= π Csc[π s] Gamma[s] In[5]:= MellinTransform[g[x], x, s]MellinTransform[1 / (x + 1), x, s] Out[5]= π Csc[π s] Gamma[s] ``` --- ``MellinTransform`` is related to ``FourierTransform`` by $\sqrt{2 \pi } \left(\mathcal{F}_x\left[f\left(e^x\right)\right](s)\right)=\mathcal{M}_x[f(x)](i s)$ : ```wl In[1]:= f[x_] := UnitBox[x - 1] In[2]:= Sqrt[2 π]FourierTransform[f[E ^ x], x, s] == MellinTransform[f[x], x, I s]//FullSimplify Out[2]= True ``` ### Possible Issues (1) Different expressions may have the same ``MellinTransform`` : ```wl In[1]:= MellinTransform[E ^ (-x), x, s] Out[1]= Gamma[s] In[2]:= MellinTransform[E ^ (-x) - 1 + x, x, s] Out[2]= Gamma[s] ``` The transforms in these examples have different regions of convergence: ```wl In[3]:= mt1 = MellinTransform[E ^ (-x), x, s, GenerateConditions -> True] Out[3]= ConditionalExpression[Gamma[s], Re[s] > 0] In[4]:= mt2 = MellinTransform[E ^ (-x) - 1 + x, x, s, GenerateConditions -> True] Out[4]= ConditionalExpression[Gamma[s], -2 < Re[s] < -1] ``` With region of convergence given, the inverse transform gives back the input: ```wl In[5]:= InverseMellinTransform[mt1, s, x] Out[5]= E^-x In[6]:= InverseMellinTransform[mt2, s, x] Out[6]= -1 + E^-x + x ``` ### Neat Examples (1) Create a table of basic Mellin transforms: ```wl In[1]:= flist = {E ^ (-a x), HeavisideTheta[x - a]x ^ b, DiracDelta[x - a], 1 / (1 + x), 1 / (1 + x ^ 2), Log[1 + x], Sin[x], Cos[x], E ^ (-x ^ 2), 1 / (E ^ x ^ (-1) x), BesselJ[n, x], BesselK[n, x]}; In[2]:= Grid[Prepend[{#, MellinTransform[#1, x, s]}& /@ flist, {f[x], MellinTransform[f[s], x, s]}], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{cc} f(x) & \mathcal{M}_x[f(s)](s) \\ e^{-a x} & a^{-s} \Gamma (s) \\ x^b \theta (x-a) & -\frac{a^{b+s}}{b+s} \\ \delta (x-a) & a^{s-1} \\ \frac{1}{x+1} & \pi \csc (\pi s) \\ \frac{1}{x^2+1} & \frac{1}{2} \pi \csc \left(\fra ... s) \\ J_n(x) & \frac{2^{s-1} \Gamma \left(\frac{n}{2}+\frac{s}{2}\right)}{\Gamma \left(\frac{n}{2}-\frac{s}{2}+1\right)} \\ K_n(x) & 2^{s-2} \Gamma \left(\frac{s}{2}-\frac{n}{2}\right) \Gamma \left(\frac{n}{2}+\frac{s}{2}\right) \\ \end{array}$$ ``` ## See Also * [`InverseMellinTransform`](https://reference.wolfram.com/language/ref/InverseMellinTransform.en.md) * [`MellinConvolve`](https://reference.wolfram.com/language/ref/MellinConvolve.en.md) * [`LaplaceTransform`](https://reference.wolfram.com/language/ref/LaplaceTransform.en.md) * [`BilateralLaplaceTransform`](https://reference.wolfram.com/language/ref/BilateralLaplaceTransform.en.md) * [`FourierTransform`](https://reference.wolfram.com/language/ref/FourierTransform.en.md) * [`HankelTransform`](https://reference.wolfram.com/language/ref/HankelTransform.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) * [`MeijerG`](https://reference.wolfram.com/language/ref/MeijerG.en.md) * [`FoxH`](https://reference.wolfram.com/language/ref/FoxH.en.md) * [`Gamma`](https://reference.wolfram.com/language/ref/Gamma.en.md) * [`MeijerGReduce`](https://reference.wolfram.com/language/ref/MeijerGReduce.en.md) * [`FoxHReduce`](https://reference.wolfram.com/language/ref/FoxHReduce.en.md) * [`Asymptotic`](https://reference.wolfram.com/language/ref/Asymptotic.en.md) ## Related Guides * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) * [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.en.md) ## History * [Introduced in 2016 (11.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn110.en.md)