# Tag Info

14

This isn't necessarily how these functions are implemented, but MathematicalFunctionData gives a way to access definitions that are equivalent to the ones Mathematica uses. (* There are a total of 348 functions to choose from *) Length[functions = MathematicalFunctionData[]] 348 functions[[1]]["Definition"] {Function[{\[FormalX]}, Inactivate[ ...

9

Integrating by parts we have: F[t_] := Exp[-a*t]; G[t_] := Log[t]*Log[1 + t]; HoldForm[Integrate[F[t]*G'[t], t] = F[t]*G[t] - Integrate[F'[t]*G[t], t]] integral = Integrate[F[t]*G'[t], t] == F[t]*G[t] - Integrate[F'[t]*G[t], t] First@Expand@Solve[integral, \[Integral]E^(-a t) Log[t] Log[1 + t] \[DifferentialD]t] \int F(t) G'(t) \, dt=F(t) G(t)-\int ...

4

In Mathematica notation \$Assumptions = ϕ ∈ Reals F[ϕ_, m_] := Integrate[1/Sqrt[1 - m Sin[θ]^2], {θ, 0, ϕ}] Plot[F[ϕ, -1], {ϕ, - Pi, π}, PlotStyle -> Red] In Maple notation F[\[Phi]_, k_] := Integrate[1/Sqrt[1 - k^2 Sin[\[Theta]]^2], {\[Theta], 0, \[Phi]}] Plot[F[\[Phi], I], {\[Phi], -2 Pi, 2 \[Pi]}] In maple

3

Analysis of the error (bug?) We can see from the trace below that the second limit, which carries out a ratio test for the product, mistakenly yields -17 (which would indicate divergence, if correct). Trace[ NProduct[(n^2)!/stirling[n^2], {n, 1, Infinity}], _Limit, TraceInternal -> True, TraceForward -> True] There might have been some ...

3

n Binomial[n - 1, k - 1] == k Binomial[n, k] // FullSimplify True

2

the culprit here is AccuracyGoal: a = 3; c = 6; d = 0.00033; b = 2; NIntegrate[ x^3 (SphericalBesselJ[0, b x] + SphericalBesselJ[2, b x])/(4 d^2 x^2 + 9)^6 1/ 2 E^(-(1/2) x^2 - I x c - 1/2 a^2) (1 + E^(2 I x c) - I E^(2 I x c) Erfi[(x - I c)/Sqrt[2]] - I Erfi[(x + I c)/Sqrt[2]]), {x, 0, 8}, MaxRecursion -> 22, AccuracyGoal ...

2

You're not going to be able to find all the roots, because there are an infinite number of them. But you can use FindRoot directly to find any subset within a range. x = 3; eqn = BesselY[1, b] BesselJ[1, b x] - BesselJ[1, b] BesselY[1, b x] == 0; sol = FindRoot[eqn, {b, #}] & /@ Range[20] Here are the first few: Sort[DeleteDuplicates[sol[[All, 1, ...

2

x = 3; f[b_] = BesselY[1, b] BesselJ[1, b x] - BesselJ[1, b] BesselY[1, b x] ; FindInstance[{f[b]==0, 0 <= b <= 10}, b, Reals, 7] sol = b /. % // N (* {1.63562, 3.17884, 4.73809, 6.30272, 7.86971, 9.43793} *) There are only 6 real roots. You also can do it with Solve or Reduce: Solve[{f[b]==0, 0 <= b <= 10}, b, Reals] Plot[f[b], {b, 0, ...

2

EDIT I have now confirmed my closed form expression numerically. This was possible by helping Mathematica to calculate the numerical value of mixed partial derivatives of HypergeometricU[a,b,z] with respect to a and b. Original post We derive a closed form by another procedure. The procedure seems to be valid but Mathematica has difficulties with the ...

1

Normal@Series[BesselK[1, r Λ]/BesselK[1, Λ], {r, 0, 1}, Assumptions -> Λ > 0] gives You can also use Simplify@Normal@Series[BesselK[1, r Λ]/BesselK[1, Λ], {r, 0, 1}, Assumptions -> Λ > 0] or Simplify[Normal@ Series[BesselK[1, r Λ]/ BesselK[1, Λ], {r, 0, 1}], Assumptions -> Λ > 0] or Assuming[{ Λ > 0}, ...

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