For fixed real $\nu$ as $z\to+\infty$, the <Modified Bessel function of the first kind> and <Modified Bessel function of the second kind> have <asymptotic expansions>
$$
I_\nu(z)\sim\frac{e^z}{\sqrt{2\pi z}}\sum_{j=0}^\infty\frac{(-1)^ja_j(\nu)}{z^j},\qquad
K_\nu(z)\sim\sqrt{\frac\pi{2z}}e^{-z}\sum_{j=0}^\infty\frac{a_j(\nu)}{z^j},\qquad
a_0=1,\quad a_{j+1}=\frac{4\nu^2-(2j+1)^2}{8(j+1)}a_j.
$$
The expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a <differential equation>. These formulas are documented in https://dlmf.nist.gov/10.40[NIST DLMF, equations 10.40.1–2].

Large-argument asymptotic expansion of a modified Bessel function

Put $q=\mu^2-1/4$ and $S=\log y$. The <differential equation> becomes the <Riccati equation> $S''+(S')^2=1/4+q/x^2$. The inverse-power exponential hierarchy begins with
$$
(S_0')^2=\frac14,\qquad2S_0'S_1'+S_0''=0,
$$
so choose $S_0=\varepsilon x/2$, $S_1=0$ with $\varepsilon=\pm1$; additive constants specify the overall normalization. At order $x^{-2}$, $2S_0'S_2'=q/x^2$, and at subsequent orders
$$
2S_0'S_j'+S_{j-1}''+\sum_{r=2}^{j-2}S_r'S_{j-r}'=0,\qquad j\geq3.
$$
These equations are formally equivalent coefficient by coefficient; the order-zero transport term happens to be constant, so its chosen zero value is not interpreted as a literal denominator in a successive-ratio hypothesis. The ansatz $S_j=c_jx^{-(j-1)}$ is consistent for $j\geq2$, with
$$
\boxed{c_2=-\varepsilon q,\qquad
c_j=\varepsilon\left[(j-2)c_{j-1}+\frac1{j-1}\sum_{r=2}^{j-2}(r-1)(j-r-1)c_rc_{j-r}\right],\quad j\geq3.}
$$
Empty sums are zero. In particular $c_3=-q$, which agrees with direct expansion of the <Riccati equation>.

Exponentiating the inverse-power series gives $y_\varepsilon\sim e^{\varepsilon x/2}\sum_{j\geq0}A_j^\varepsilon x^{-j}$ with $A_0=1$. Substitute this amplitude series into $u''+\varepsilon u'=qx^{-2}u$ to obtain
$$
\boxed{A_{j+1}^\varepsilon=\frac{\varepsilon[j(j+1)-q]}{j+1}A_j^\varepsilon,\qquad
A_1^\varepsilon=-\varepsilon q,\quad A_2^\varepsilon=\frac{q(q-2)}2.}
$$
Formal matching alone does not prove existence of actual solutions with these asymptotics. Here actual solutions are obtained from the <Modified Bessel differential equation> and the <large-argument asymptotic expansion of a modified Bessel function>: substituting $y=\sqrt x\,w(x/2)$ yields solutions $\sqrt\pi\sqrt x\,I_\mu(x/2)$ and $\sqrt{x/\pi}\,K_\mu(x/2)$, using the <Modified Bessel function of the first kind> and <Modified Bessel function of the second kind>, whose positive-real-axis <asymptotic expansions> have precisely these normalized growing and decaying series. They are <linearly independent>. The series need not converge; it is an <asymptotic expansion> with a remainder after each fixed truncation. When $q=j(j+1)$ the amplitude recurrence terminates.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 30E / Solution

{c}
{title2=$I_\nu(z)$}

The solution of the <Modified Bessel differential equation> with <power series> $I_\nu(z)=\sum_{j\geq0}(z/2)^{2j+\nu}/[j!\Gamma(j+\nu+1)]$, interpreted through the reciprocal <gamma function> at its zeros. For fixed real order it grows exponentially along the positive real axis.

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 Incoming links: Modified Bessel function of the first kind