91Ó°ÊÓ

Recurrence formulas that relate Bessel functions of different orders are important in theory and in applications.

\(\frac{d}{{dx}}\left( {{x^{ - v}}{J_v}(x)} \right) = - {x^{ - v}}{J_{v + 1}}(x)\)… (1)

\(\frac{d}{{dx}}\left( {{x^v}{J_v}(x)} \right) = {x^v}{J_{v - 1}}(x)\) … (2)

Substitute the value\(v = 0\)in the equation (1).

\(\begin{array}{c}\frac{d}{{dx}}\left( {{x^0}{J_0}(x)} \right) = - {x^0}{J_{0 + 1}}(x)\\\frac{d}{{dx}}\left( {{J_0}(x)} \right) = {J_1}(x)\end{array}\)

\(J_0^'(x) = - {J_1}(x)\)… (3)

Substitute the value\(v = 0\)in the equation (2).

\(\begin{array}{c}\frac{d}{{dx}}\left( {{x^0}{J_0}(x)} \right) = {x^0}{J_{0 - 1}}(x)\\\frac{d}{{dx}}\left( {{J_0}(x)} \right) = {J_{ - 1}}(x)\end{array}\)

\(J_0^'(x) = {J_{ - 1}}(x)\)… (4)

From (3) and (4) yields the result,

\(J_0^'(x) = - {J_1}(x) = {J_{ - 1}}(x)\)