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Mathematics LibreTexts

10.5: Properties of Bessel functions

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Bessel functions have many interesting properties: J0(0)=1,Jν(x)=0(if ν>0),Jn(x)=(1)nJn(x),ddx[xνJν(x)]=xνJν+1(x),ddx[xνJν(x)]=xνJν1(x),ddx[Jν(x)]=12[Jν1(x)Jν+1(x)],xJν+1(x)=2νJν(x)xJν1(x),xνJν+1(x)dx=xνJν(x)+C,xνJν1(x)dx=xνJν(x)+C.

Let me prove a few of these. First notice from the definition that Jn(x) is even or odd if n is even or odd,

Jn(x)=k=0(1)kk!(n+k)!(x2)n+2k.

Substituting x=0 in the definition of the Bessel function gives 0 if ν>0, since in that case we have the sum of positive powers of 0, which are all equally zero.

Let’s look at Jn:

Jn(x)=k=0(1)kk!Γ(n+k+1)!(x2)n+2k=k=n(1)kk!Γ(n+k+1)!(x2)n+2k=l=0(1)l+n(l+n)!l!(x2)n+2l=(1)nJn(x).

Here we have used the fact that since Γ(l)=±, 1/Γ(l)=0 [this can also be proven by defining a recurrence relation for 1/Γ(l)]. Furthermore we changed summation variables to l=n+k.

The next one:

ddx[xνJν(x)]=2νddx{k=0(1)kk!Γ(ν+k+1)(x2)2k}=2νk=1(1)k(k1)!Γ(ν+k+1)(x2)2k1=2νl=0(1)l(l)!Γ(ν+l+2)(x2)2l+1=2νl=0(1)l(l)!Γ(ν+1+l+1)(x2)2l+1=xνl=0(1)l(l)!Γ(ν+1+l+1)(x2)2l+ν+1=xνJν+1(x).

Similarly ddx[xνJν(x)]=xνJν1(x).

The next relation can be obtained by evaluating the derivatives in the two equations above, and solving for Jν(x):

xνJν(x)νxν1Jν(x)=xνJν+1(x),xνJν(x)+νxν1Jν(x)=xνJν1(x).

Multiply the first equation by xν and the second one by xν and add:

2ν1xJν(x)=Jν+1(x)+Jν1(x).

After rearrangement of terms this leads to the desired expression.

Eliminating Jν between the equations gives (same multiplication, take difference instead) 2Jν(x)=Jν+1(x)+Jν1(x).

Integrating the differential relations leads to the integral relations.

Bessel function are an inexhaustible subject – there are always more useful properties than one knows. In mathematical physics one often uses specialist books.


This page titled 10.5: Properties of Bessel functions is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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