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10.5 Bessel functions have many interesting properties:
\eqalignno{
{J}_{0}(0) & = 1, &\text{(10.31)}
\cr
{J}_{ν}(x) & = 0\quad \text{(if $ν > 0$),} &\text{(10.32)}
\cr
{J}_{−n}(x) & = {(−1)}^{n}{J}_{
n}(x), &\text{(10.33)}
\cr
{d\over
dx}\left [{x}^{−ν}{J}_{
ν}(x)\right ] & = −{x}^{−ν}{J}_{
ν+1}(x), &\text{(10.34)}
\cr
{d\over
dx}\left [{x}^{ν}{J}_{
ν}(x)\right ] & = {x}^{ν}{J}_{
ν−1}(x), &\text{(10.35)}
\cr
{d\over
dx}\left [{J}_{ν}(x)\right ] & = {1\over
2}\left [{J}_{ν−1}(x) − {J}_{ν+1}(x)\right ], &\text{(10.36)}
\cr
x{J}_{ν+1}(x) & = 2ν{J}_{ν}(x) − x{J}_{ν−1}(x), &\text{(10.37)}
\cr
\mathop{\mathop{\mathop{∫
}\nolimits }}\nolimits {x}^{−ν}{J}_{
ν+1}(x)\kern 1.66702pt dx & = −{x}^{−ν}{J}_{
ν}(x) + C, &\text{(10.38)}
\cr
\mathop{\mathop{\mathop{∫
}\nolimits }}\nolimits {x}^{ν}{J}_{
ν−1}(x)\kern 1.66702pt dx & = {x}^{ν}{J}_{
ν}(x) + C. &\text{(10.39)}
}
Let me prove a few of these. First notice from the definition that
{J}_{n}(x) is even or
odd if n is
even or odd,
{J}_{n}(x) ={ \mathop{∑
}}_{k=0}^{∞} {{(−1)}^{k}\over
k!(n + k)!}{\left ({x\over
2}\right )}^{n+2k}.
(10.40)
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} :
\begin{eqnarray}{
J}_{−n}(x)& =& {\mathop{∑
}}_{k=0}^{∞} {{(−1)}^{k}\over
k!Γ(−n + k + 1)!}{\left ({x\over
2}\right )}^{n+2k} %&
\\
& =& {\mathop{∑
}}_{k=n}^{∞} {{(−1)}^{k}\over
k!Γ(−n + k + 1)!}{\left ({x\over
2}\right )}^{−n+2k}%&
\\
& =& {\mathop{∑
}}_{l=0}^{∞}{{(−1)}^{l+n}\over
(l + n)!l!}{\left ({x\over
2}\right )}^{n+2l} %&
\\
& =& {(−1)}^{n}{J}_{
n}(x). %&(10.41) \\
\end{eqnarray}
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:
\begin{eqnarray}
{d\over
dx}\left [{x}^{−ν}{J}_{
ν}(x)\right ]& =& {2}^{−ν} {d\over
dx}\left \{{\mathop{∑
}}_{k=0}^{∞} {{(−1)}^{k}\over
k!Γ(ν + k + 1)}{\left ({x\over
2}\right )}^{2k}\right \} %&
\\
& =& {2}^{−ν}{ \mathop{∑
}}_{k=1}^{∞} {{(−1)}^{k}\over
(k − 1)!Γ(ν + k + 1)}{\left ({x\over
2}\right )}^{2k−1} %&
\\
& =& −{2}^{−ν}{ \mathop{∑
}}_{l=0}^{∞} {{(−1)}^{l}\over
(l)!Γ(ν + l + 2)}{\left ({x\over
2}\right )}^{2l+1} %&
\\
& =& −{2}^{−ν}{ \mathop{∑
}}_{l=0}^{∞} {{(−1)}^{l}\over
(l)!Γ(ν + 1 + l + 1)}{\left ({x\over
2}\right )}^{2l+1} %&
\\
& =& −{x}^{−ν}{ \mathop{∑
}}_{l=0}^{∞} {{(−1)}^{l}\over
(l)!Γ(ν + 1 + l + 1)}{\left ({x\over
2}\right )}^{2l+ν+1}%&
\\
& =& −{x}^{−ν}{J}_{
ν+1}(x). %&(10.42) \\
\end{eqnarray}
Similarly
\begin{eqnarray}
{d\over
dx}\left [{x}^{ν}{J}_{
ν}(x)\right ]& =& {x}^{ν}{J}_{
ν−1}(x).%&(10.43) \\
\end{eqnarray}
The next relation can be obtained by evaluating the derivatives in the two equations above, and solving for
{J}_{ν}(x) :
\begin{eqnarray}{
x}^{−ν}{J'}_{
ν}(x) − ν{x}^{−ν−1}{J}_{
ν}(x)& =& −{x}^{−ν}{J}_{
ν+1}(x),%&(10.44)
\\
{x}^{ν}{J}_{
ν}(x) + ν{x}^{ν−1}{J}_{
ν}(x)& =& {x}^{ν}{J}_{
ν−1}(x). %&(10.45) \\
\end{eqnarray}
Multiply the first equation by {x}^{ν}
and the second one by {x}^{−ν}
and add:
\begin{eqnarray}
−2ν {1\over
x}{J}_{ν}(x) = −{J}_{ν+1}(x) + {J}_{ν−1}(x).& & %&(10.46) \\
\end{eqnarray}
After rearrangement of terms this leads to the desired expression.
Eliminating {J}_{ν}
between the equations gives (same multiplication, take difference instead)
\begin{eqnarray}
2{J'}_{ν}(x)& =& {J}_{ν+1}(x) + {J}_{ν−1}(x).%&(10.47) \\
\end{eqnarray}
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.