--- title: "AppellF3" language: "en" type: "Symbol" summary: "AppellF3[a1, a2, b1, b2, c, x, y] is the Appell hypergeometric function of two variables F3 (a; b1, b2; c1, c2; x, y)." keywords: - Appell hypergeometric function - hypergeometric functions - multivariate hypergeometric function canonical_url: "https://reference.wolfram.com/language/ref/AppellF3.html" source: "Wolfram Language Documentation" related_guides: - title: "Hypergeometric Functions" link: "https://reference.wolfram.com/language/guide/HypergeometricFunctions.en.md" related_functions: - title: "AppellF1" link: "https://reference.wolfram.com/language/ref/AppellF1.en.md" - title: "AppellF2" link: "https://reference.wolfram.com/language/ref/AppellF2.en.md" - title: "AppellF4" link: "https://reference.wolfram.com/language/ref/AppellF4.en.md" - title: "Hypergeometric2F1" link: "https://reference.wolfram.com/language/ref/Hypergeometric2F1.en.md" - title: "Gamma" link: "https://reference.wolfram.com/language/ref/Gamma.en.md" - title: "Pochhammer" link: "https://reference.wolfram.com/language/ref/Pochhammer.en.md" --- [EXPERIMENTAL] # AppellF3 AppellF3[a1, a2, b1, b2, c, x, y] is the Appell hypergeometric function of two variables $F_3\left(a;b_1,b_2;c_1,c_2;x,y\right)$. ## Details * ``AppellF3`` belongs to the family of Appell functions that generalize the hypergeometric series and solves the system of Horn PDEs with polynomial coefficients. * Mathematical function, suitable for both symbolic and numerical manipulation. * $F_3\left(a_1,a_2;b_1,b_2;c;x,y\right)$ has a primary definition through the hypergeometric series $\sum _{m=0}^{\infty } \sum _{n=0}^{\infty } \frac{\left(a_1\right)_m\left(a_2\right)_n\left(b_1\right)_m\left(b_2\right)_n}{(c)_{m+n}m!n!}x^my^n$, which is convergent inside the region $\max (| x| ,| y| )<1$. * The region of convergence of the Appell F3 series for real values of its arguments is the following: [image] * In general, $F_3\left(a_1,a_2;b_1,b_2;c;x,y\right)$ satisfies the following Horn PDE system »: $x (1-x) \frac{\partial ^2f}{\partial x^2}+y \frac{\partial ^2f}{\partial x\partial y}+\left(c-\left(a_1+b_1+1\right)x\right)\frac{\partial f}{\partial x}-a_1b_1 f=0$$ $$y (1-y) \frac{\partial ^2f}{\partial y^2}+x \frac{\partial ^2f}{\partial x\partial y}+\left(c-\left(a_2+b_2+1\right)y\right)\frac{\partial f}{\partial y}-a_2b_2 f=0$ * $F_3\left(a_1,a_2;b_1,b_2;c;x,y\right)$ reduces to $_2F_1(a,b;c;z)$ when $x=0$ or $y=0$. * For certain special arguments, ``AppellF3`` automatically evaluates to exact values. * ``AppellF3`` can be evaluated to arbitrary numerical precision. ## Examples (26) ### Basic Examples (7) Evaluate numerically: ```wl In[1]:= AppellF3[2, 1, 3, 4, 5, 0.7, 0.3] Out[1]= 4.39557 ``` --- The defining sum: ```wl In[1]:= Sum[x^m y^n(( Pochhammer[a1, m]Pochhammer[a2, n]Pochhammer[b1, m] Pochhammer[b2, n]) / (Pochhammer[c, m + n]m! n!)), {m, 0, Infinity}, {n, 0, Infinity}] Out[1]= AppellF3[a1, a2, b1, b2, c, x, y] ``` --- Plot over a subset of the reals: ```wl In[1]:= Plot[AppellF3[1, 2, 3, 6, 1 / 5, -1 / 3, y], {y, -3 / 4, 1 / 5}] Out[1]= [image] ``` --- Plot over a subset of the complexes: ```wl In[1]:= Plot[Abs[AppellF3[1 / 2, 1 / 3, 1 / 4, 1 / 5, 1 / 6, 1 / 7, z I]], {z, -3 / 4, 3 / 4}] Out[1]= [image] ``` --- Plot a family of ``AppellF3`` functions: ```wl In[1]:= Plot[Table[Abs[AppellF3[1 / 2, 1 / 3, 1 / 4, 1 / 5, 1 / 6, 1 / 2 + n / 5 I, 1 / 2 + z I]], {n, -3, 3}]//Evaluate, {z, -9 / 10, 1}] Out[1]= [image] ``` --- Series expansion at the origin: ```wl In[1]:= Series[AppellF3[1, 1, 1, 1, 1, 4, x], {x, 0, 3}] Out[1]= SeriesData[x, 0, {Rational[-1, 3], (Complex[0, Rational[-1, 4]])*(Pi + (Complex[0, -1])*Log[3]), Rational[1, 8]*(4 + (Complex[0, -3])*Pi - 3*Log[3]), Rational[3, 64]*(20 + (Complex[0, -9])*Pi - 9*Log[3])}, 0, 4, 1] ``` --- ``TraditionalForm`` formatting: ```wl In[1]:= AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, y]//TraditionalForm Out[1]//TraditionalForm= $$F_3\left(a_1,a_2;b_1,b_2;c;x,y\right)$$ ``` ### Scope (17) #### Numerical Evaluation (6) Evaluate numerically: ```wl In[1]:= AppellF3[3, 2, 1, 2, 3, 1 / 3, 0.2] Out[1]= 1.95293 In[2]:= AppellF3[-2, -1, -2, 5, 2 / 3 + I, 0.4, 0.3] Out[2]= 1.02589  + 0.888865 I ``` --- Evaluate to high precision: ```wl In[1]:= N[AppellF3[3, 2, 1, 2, 3, 1 / 3, 2 / 5], 10] Out[1]= 2.725409214 ``` The precision of the output tracks the precision of the input: ```wl In[2]:= AppellF3[3, 2, 1, 2, 3, 1 / 3, 0.400000000000000000001] Out[2]= 2.72540921450408658070 ``` --- Complex number inputs: ```wl In[1]:= AppellF3[I, 1, 1 + I, 3.2, 1 / 2, 0.5, 0.2 + 0.5 I] Out[1]= -3.63153 - 0.760372 I ``` --- Evaluate ``AppellF3`` efficiently at high precision: ```wl In[1]:= AppellF3[3, 2, 1, 2, 1 / 3, 1 / 7, 1 / 5`100]//Timing Out[1]= {0.165667, 9.752192846585162639544050084260060081941593341356626122811062126085239631548698947450609236257242846} In[2]:= AppellF3[3, 2, 1, 2, 7, 1 / 3, -1 / 5`100 + I];//Timing Out[2]= {0.565133, Null} ``` --- Compute average-case statistical intervals using ``Around``: ```wl In[1]:= AppellF3[ 1 / 2, 3 / 2, -4, 5, 1 / 2, 0, Around[2.1, 0.01]] Out[1]= Around[11.233031208070166, 0.6148133258862857] ``` --- Compute the elementwise values of an array: ```wl In[1]:= AppellF3[1 / 2, 3 / 2, -4, 5, 1 / 2, 0, {{1 / 2, 2}, {2, 1 / 2}}] Out[1]= {{352, 19}, {19, 352}} ``` Or compute the matrix ``AppellF3`` function using ``MatrixFunction``: ```wl In[2]:= MatrixFunction[AppellF3[1 / 2, 3 / 2, -4, 5, 1 / 2, 0, #]&, {{1 / 2, 2}, {2, 1 / 2}}]//FullSimplify Out[2]= {{(458336/455625), (481664/455625)}, {(481664/455625), (458336/455625)}} ``` #### Specific Values (3) Values at fixed points: ```wl In[1]:= AppellF3[1 / 2, 1 / 3, 1 / 2, 1, 3, 0, 2] Out[1]= -(3/20) (-7 + 3 (-1)^2 / 3) In[2]:= AppellF3[2, 1 / 3, 1 / 4, 1, 4, 0, 2] Out[2]= -(9/320) (-41 + 9 (-1)^2 / 3) ``` --- Simplify to ``Hypergeometric2F1`` functions: ```wl In[1]:= AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, 0] Out[1]= Hypergeometric2F1[Subscript[a, 1], Subscript[b, 1], c, x] In[2]:= AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, 0, y] Out[2]= Hypergeometric2F1[Subscript[a, 2], Subscript[b, 2], c, y] ``` --- Value at zero: ```wl In[1]:= AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, 0, 0] Out[1]= 1 ``` #### Visualization (3) Plot the ``AppellF3`` function for various parameters: ```wl In[1]:= Plot[{AppellF3[1, 2, 3, 6, 1 / 5, x, 1 / 30], AppellF3[2, 1, 3, 1 / 3, 2, x, 0], AppellF3[2, 1, 3, 1 / 3, 2, x, -1 / 2]}, {x, -1 / 2, 1 / 5}] Out[1]= [image] ``` --- Plot ``AppellF3`` as a function of its second parameter $y$ : ```wl In[1]:= Plot[{AppellF3[1, 2, 3, 6, 1 / 5, -1 / 2, y], AppellF3[1, 2, 3, 6, 1 / 5, -1 / 5, y], AppellF3[1, 2, 3, 6, 1 / 5, -1 / 100, y]}, {y, -1 / 2, 1 / 5}] Out[1]= [image] ``` --- Plot the real part of $F_3(2,1,3,1/3,2,0,z)$ : ```wl In[1]:= ComplexContourPlot[Re[AppellF3[2, 1, 3, 1 / 3, 2, 0, z]], {z, -5 - 5I, 5 + 5I}] Out[1]= [image] ``` Plot the imaginary part of $F_3(2,1,3,1/3,2,0,z)$ : ```wl In[2]:= ComplexContourPlot[Im[AppellF3[2, 1, 3, 1 / 3, 2, 0, z]], {z, -5 - 5I, 5 + 5I}] Out[2]= [image] ``` #### Differentiation (4) First derivative with respect to ``x`` : ```wl In[1]:= D[AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, y], x] Out[1]= (AppellF3[1 + Subscript[a, 1], Subscript[a, 2], 1 + Subscript[b, 1], Subscript[b, 2], 1 + c, x, y] Subscript[a, 1] Subscript[b, 1]/c) ``` --- First derivative with respect to ``y`` : ```wl In[1]:= D[AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, y], y] Out[1]= (AppellF3[Subscript[a, 1], 1 + Subscript[a, 2], Subscript[b, 1], 1 + Subscript[b, 2], 1 + c, x, y] Subscript[a, 2] Subscript[b, 2]/c) ``` --- Higher derivatives with respect to ``y`` : ```wl In[1]:= Table[D[AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, y], {y, k}], {k, 1, 3}]//FullSimplify Out[1]= {(AppellF3[Subscript[a, 1], 1 + Subscript[a, 2], Subscript[b, 1], 1 + Subscript[b, 2], 1 + c, x, y] Subscript[a, 2] Subscript[b, 2]/c), (AppellF3[Subscript[a, 1], 2 + Subscript[a, 2], Subscript[b, 1], 2 + Subscript[b, 2], 2 + c, x, y] Subscript[a, ... 1 + c)), (AppellF3[Subscript[a, 1], 3 + Subscript[a, 2], Subscript[b, 1], 3 + Subscript[b, 2], 3 + c, x, y] Subscript[a, 2] (1 + Subscript[a, 2]) (2 + Subscript[a, 2]) Subscript[b, 2] (1 + Subscript[b, 2]) (2 + Subscript[b, 2])/c (1 + c) (2 + c))} ``` Plot the higher derivatives with respect to ``y`` when ``a1 = a2 = 2``, `` b1 = b2 = 5``, `` c = 1 / 2`` and ``x = 1 / 5`` : ```wl In[2]:= Plot[Evaluate[% /. {Subscript[a, 1] -> 2, Subscript[a, 2] -> 2, Subscript[b, 1] -> 5, Subscript[b, 2] -> 5, c -> 1 / 2, x -> 1 / 5}], {y, -1 / 2, 1 / 5}, Rule[...]] Out[2]= [image] ``` --- Formula for the $n$$$^{\text{th}}$$ derivative with respect to ``y`` : ```wl In[1]:= D[AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, y], {y, n}] Out[1]= (AppellF3[Subscript[a, 1], n + Subscript[a, 2], Subscript[b, 1], n + Subscript[b, 2], c + n, x, y] Pochhammer[Subscript[a, 2], n] Pochhammer[Subscript[b, 2], n]/Pochhammer[c, n]) ``` #### Series Expansions (1) Find the Taylor expansion using ``Series``: ```wl In[1]:= Series[AppellF3[Subscript[a, 1], Subscript[a, 2], Subscript[b, 1], Subscript[b, 2], c, x, y], {x, 0, 2}]//Normal//FullSimplify Out[1]= Hypergeometric2F1[Subscript[a, 2], Subscript[b, 2], c, y] + x Gamma[c] Hypergeometric2F1Regularized[Subscript[a, 2], Subscript[b, 2], 1 + c, y] Subscript[a, 1] Subscript[b, 1] + (1/2) x^2 Gamma[c] Hypergeometric2F1Regularized[Subscript[a, 2], Subscript[b, 2], 2 + c, y] Subscript[a, 1] (1 + Subscript[a, 1]) Subscript[b, 1] (1 + Subscript[b, 1]) ``` Plots of the first three approximations around $x=0$ : ```wl In[2]:= terms = Normal@Table[Series[AppellF3[1, 2, 3, 6, 1 / 5, x, .3], {x, 0, m}], {m, 1, 5, 2}]; Plot[{AppellF3[1, 2, 3, 6, 1 / 5, x, .3], terms}//Evaluate, {x, -9 / 10, 1 / 2}] Out[2]= [image] ``` ### Applications (1) The Appell function $F_3\left(a_1,a_2;b_1,b_2;c;x,y\right)$ solves the following system of PDEs with polynomial coefficients: ```wl In[1]:= pde = {x (1 - x) f^(2, 0)[x, y] + y f^(1, 1)[x, y] + (c - (Subscript[a, 1] + Subscript[b, 1] + 1)x)f^(1, 0)[x, y] - Subscript[a, 1] Subscript[b, 1] f[x, y] == 0, y (1 - y) f^(0, 2)[x, y] + x f^(1, 1)[x, y] + (c - (Subscript[a, 2] + Subscript[b, 2] + 1)y)f^(0, 1)[x, y] - Subscript[a, 2] Subscript[b, 2] f[x, y] == 0}; ``` Check that $F_3(1,2;3,6;1/5;x,y)$ is a solution: ```wl In[2]:= (pde /. {Subscript[a, 1] -> 1, Subscript[a, 2] -> 2, Subscript[b, 1] -> 3, Subscript[b, 2] -> 6, c -> 1 / 5}) /. f -> Function[{x, y}, AppellF3[1, 2, 3, 6, 1 / 5, x, y]]; In[3]:= % /. Thread[{x, y} -> RandomReal[{-1 / 2, 1 / 2}, {2}, WorkingPrecision -> 50]] Out[3]= {True, True} ``` ### Neat Examples (1) Many elementary and special functions are special cases of ``AppellF3`` : ```wl In[1]:= funclist = Inactivate[...]; In[2]:= Grid[...]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{|c|c|} \hline \text{Special case of AppellF3} & \text{Simpler function} \\ \hline F_3\left(a_1,a_2;b_1,b_2;c;x,0\right) & \, _2F_1\left(a_1,b_1;c;x\right) \\ \hline F_3\left(a_1,a_2;b_1,b_2;c;0,y\right) & \, _2F_1\left(a_2,b_2;c; ... \right) & \, _2F_1\left(a_1,a_1+\frac{1}{2};2 a_1;x\right) \\ \hline F_3\left(a_1,\frac{1}{2};b_1,\frac{1}{2};1;0,y\right) & \frac{2 K(y)}{\pi } \\ \hline F_3\left(a_1,1;b_1,\frac{1}{2};1;0,y\right) & \frac{1}{\sqrt{1-y}} \\ \hline \end{array}$$ ``` ## See Also * [`AppellF1`](https://reference.wolfram.com/language/ref/AppellF1.en.md) * [`AppellF2`](https://reference.wolfram.com/language/ref/AppellF2.en.md) * [`AppellF4`](https://reference.wolfram.com/language/ref/AppellF4.en.md) * [`Hypergeometric2F1`](https://reference.wolfram.com/language/ref/Hypergeometric2F1.en.md) * [`Gamma`](https://reference.wolfram.com/language/ref/Gamma.en.md) * [`Pochhammer`](https://reference.wolfram.com/language/ref/Pochhammer.en.md) ## Related Guides * [Hypergeometric Functions](https://reference.wolfram.com/language/guide/HypergeometricFunctions.en.md) ## Related Links * [MathWorld](http://mathworld.wolfram.com/AppellHypergeometricFunction.html) ## History * [Introduced in 2023 (13.3)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn133.en.md)