# How to write a specific Bessel function in Mathematica

I want to plot the following function on Mathematica, and I gave it a go on wolframalpha.

Bessel[n,z] is the usual form, but I am not sure how to use this to compute the following plot:

$$\begin{equation} u(r,t)=\frac{\alpha J_{4}(i\sqrt{2}r)}{J_{4}\big(\frac{100*2}{\alpha}\big)}e^{-16t^2} \end{equation}$$

I tried

BesselJ[4, I Sqrt x]/BesselJ[4, 200]


But I don't know how to include the zeros defined by $$\alpha$$

Any help appreciated!

where $$\alpha$$ are the zeros of the Bessel function.

• By "zeros of the Bessel function," do you mean alpha = BesselJZero[n, k]? For n = 4 for k = 1, 2, 3,....? May 21, 2022 at 15:26
• Yes, precisely. May 21, 2022 at 15:27

Clear["Global*"]

u[x_, t_, α_] := α*BesselJ[4, I Sqrt x]/BesselJ[4, 200/α]*
E^(-16 t^2)

u[x, t, α] == -u[x, t, -α]

(* True *)

u[x, t, α] == u[x, -t, α]

(* True *)

u[x, t, α] == u[-x, t, α]

(* True *)

Manipulate[
Plot3D[u[x, t, α],
{x, -5, 5}, {t, -2, 2},
AxesLabel -> Automatic,
ClippingStyle -> None],
{{α, 1}, 0.05, 5, 0.05, Appearance -> "Labeled"}] EDIT: For α = BesselJZero[4, k]

Manipulate[
α = BesselJZero[4, k];
Plot3D[u[x, t, α], {x, -5, 5}, {t, -2, 2},
AxesLabel -> Automatic,
ClippingStyle -> None,
PlotLabel -> StringForm["α =  = ", α, α // N],
WorkingPrecision -> 15],
{{k, 1}, Range, ControlType -> SetterBar}]
` 