91Ó°ÊÓ

Let the Bessel equation be\({x^2}y'' + xy' + \left( {{x^2} - {v^2}} \right)y = 0\).A Bessel equation of the first kind,by using the coefficients\({c_{2n}}\)and\(r = v\), indicated by\({J_n}(x)\), is,

\({J_v}(x) = \sum\limits_{n = 0}^x {\frac{{{{( - 1)}^n}}}{{n!\Gamma (1 + v + n)}}} {\left( {\frac{x}{2}} \right)^{2n + v}}\)

To verify the formula, \({I_v}(x) = {i^{ - v}}{J_V}(ix)\) …(1)

Which does not involve complex variables but gives a real function. Then, the series is,

\({J_v}(x) = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!\Gamma (1 + v + n)}}} {\left( {\frac{x}{2}} \right)^{2n + v}}\)

\({J_v}(ix) = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!\Gamma (1 + v + n)}}} {\left( {\frac{{ix}}{2}} \right)^{2n + v}}\) … (2)

Substitute the equation (2) into (1) which yields,

This implies that, \({I_v}(x)\) is a real function of all values of \(n\). Hence verified.