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),J−n(x)=(−1)nJn(x),ddx[x−νJν(x)]=−x−νJν+1(x),ddx[xνJν(x)]=xνJν−1(x),ddx[Jν(x)]=12[Jν−1(x)−Jν+1(x)],xJν+1(x)=2νJν(x)−xJν−1(x),∫x−νJν+1(x)dx=−x−νJν(x)+C,∫xνJν−1(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 J−n:
J−n(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[x−νJν(x)]=2−νddx{∞∑k=0(−1)kk!Γ(ν+k+1)(x2)2k}=2−ν∞∑k=1(−1)k(k−1)!Γ(ν+k+1)(x2)2k−1=−2−ν∞∑l=0(−1)l(l)!Γ(ν+l+2)(x2)2l+1=−2−ν∞∑l=0(−1)l(l)!Γ(ν+1+l+1)(x2)2l+1=−x−ν∞∑l=0(−1)l(l)!Γ(ν+1+l+1)(x2)2l+ν+1=−x−νJν+1(x).
The next relation can be obtained by evaluating the derivatives in the two equations above, and solving for Jν(x):
x−νJ′ν(x)−νx−ν−1Jν(x)=−x−νJν+1(x),xνJν(x)+νxν−1Jν(x)=xνJν−1(x).
Multiply the first equation by xν and the second one by x−ν and add:
−2ν1xJν(x)=−Jν+1(x)+Jν−1(x).
After rearrangement of terms this leads to the desired expression.
Eliminating Jν between the equations gives (same multiplication, take difference instead) 2J′ν(x)=Jν+1(x)+Jν−1(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.