How can I inform Mathematica of an identity concerning Bessel functions?

Question

I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function.

$$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\tau)}\mathrm d\tau$$

However, when making the calculation, Mathematica does not seem to know the identity shown above and does not use it. How can I inform Mathematica of the identity so it will use it and give me results expressed in terms of Bessel functions?

Answer 1

In[1]:= $Version

Out[1]= "13.2.1 for Microsoft Windows (64-bit) (January 27, 2023)"

In[2]:= Integrate[Exp[ I x  Sin[u] - n I  x], {u, -\[Pi], \[Pi]}]

Out[2]= ConditionalExpression[2 E^(-I n x) \[Pi] BesselJ[0, Abs[x]], 
 x \[Element] Reals]

In[7]:= Series[
  2 E^(-I n x) \[Pi] BesselJ[0, Abs[x]] == 
   Integrate[Exp[ I x  Sin[u] - n I  x], {u, -\[Pi], \[Pi]}], {x, 0, 
   13}] // FullSimplify

Out[7]= True

The simple Leibniz Trick (as we see it today after the axiomatic introduction of differential forms as multilinear maps 100 years ago) can always be applied by supplying the differential dx as a symbol and write the trivial substitutin rules

In[5]:= integrate[
  Exp[ I x  Sin[u] - n I  x] du , {u, -\[Pi], \[Pi]}] /. {  
  x -> v/n ,  Sin[u] -> p, du -> dp/Sqrt[1 - p^2], u -> ArcSin[p]}

Out[5]= integrate[(dp E^(-I v + (I p v)/n))/Sqrt[
 1 - p^2], {ArcSin[p], -\[Pi], \[Pi]}]

In[7]:= E^(-I v)
  Integrate[(E^(((I p v)/n))) /Sqrt[1 - p^2], {p, -1, 1}]

Out[7]= E^(-I v) \[Pi] BesselJ[0, v/n]

keeping in mind, that the substutions are invertibly valid in intervals of monotony only.

In general, you may add known integrals from tables to your treasure trove

  Unprotect[Integrate];
    Integrate[
      Exp[ I x_Symbol*Sin[u_] - 
        I  x_Symbol (n_Symbol | n_?NumericQ)], {u_, -\[Pi], \[Pi]}] :=
     (1/(2 \[Pi]) BesselJ[n , x] /; n \[Element] Integers && n >= 0 )
    
    Protect[Integrate];

Answer 2

One method is to set the function as

J[n_, x_] := (1/(2*Pi))*Integrate[Exp[I*(x*Sin[u] - n*u)], {u, -Pi, Pi}];

As a simple use case the script

J[n_, x_] := (1/(2*Pi))*Integrate[Exp[I*(x*Sin[u] - n*u)], {u, -Pi, Pi}];
Sum[J[n, 0], {n, 0, Infinity}]

provides the result of 1, the desired result.

Since the Bessel function of the first kind is known to be useful in the calculation, based upon the description by the OP, then it would be more useful to use the series expansion, or other forms, built into Mathematica for calculations and then convert the result to the desired form.