Derive os valores limitadores para as funções esféricas modificadas de Bessel
i_{n}(x) \approx \frac{x^{n}}{(2 n+1)!!}, \quad k_{n}(x) \approx \frac{(2 n-1)!!}{x^{n+1}}, \quad x \ll 1.

Passo 1

O enunciado pede a forma assintótica para x\ll1 das funções esféricas modificadas de Bessel
i_{n}(x)=\sqrt{\frac\pi{2x}}\,I_{n+1/2}(x), \qquad k_{n}(x)=\sqrt{\frac\pi{2x}}\,K_{n+1/2}(x),
onde usaremos as expansões para I_{\nu}(x) e K_{\nu}(x) em x\to0 e as relações
\Gamma\bigl(n+\tfrac32\bigr)=\frac{(2n+1)!!}{2^{\,n+1}}\sqrt\pi, \qquad \Gamma\bigl(n+\tfrac12\bigr)=\frac{(2n-1)!!}{2^{\,n}}\sqrt\pi.

Passo 2

Para I_{\nu}(x) temos a expansão em x\to0:
I_{\nu}(x)\approx\frac{1}{\Gamma(\nu+1)}\Bigl(\frac{x}{2}\Bigr)^{\!\nu}.
Tomando \nu=n+\tfrac12,
I_{n+1/2}(x)\approx\frac{1}{\Gamma(n+\tfrac32)}\Bigl(\frac{x}{2}\Bigr)^{n+1/2}.
Logo
i_{n}(x) =\sqrt{\frac\pi{2x}}\;I_{n+1/2}(x) \approx\sqrt{\frac\pi{2x}}\;\frac{x^{\,n+1/2}}{2^{\,n+1/2}\,\Gamma(n+\tfrac32)} =\frac{\sqrt\pi}{2^{\,n+1}\,\Gamma(n+\tfrac32)}\,x^{n}.
Usando \Gamma(n+\tfrac32)=\frac{(2n+1)!!}{2^{\,n+1}}\sqrt\pi cancela-se \sqrt\pi e 2^{\,n+1}, ficando
i_{n}(x)\approx\frac{x^{n}}{(2n+1)!!}.

Passo 3

Para K_{\nu}(x) em x\to0 vale
K_{\nu}(x)\approx\frac12\,\Gamma(\nu)\,\Bigl(\frac{x}{2}\Bigr)^{\!-\nu}.
Com \nu=n+\tfrac12 obtemos
K_{n+1/2}(x)\approx\frac12\,\Gamma(n+\tfrac12)\,\Bigl(\frac{x}{2}\Bigr)^{-n-1/2}.
Então
k_{n}(x) =\sqrt{\frac\pi{2x}}\;K_{n+1/2}(x) \approx\sqrt{\frac\pi{2x}}\;\frac12\,\Gamma(n+\tfrac12)\,\frac{2^{\,n+1/2}}{x^{\,n+1/2}} =\frac{\Gamma(n+\tfrac12)\,\sqrt\pi}{2}\;x^{-n-1}.
Usando \Gamma(n+\tfrac12)=\frac{(2n-1)!!}{2^{\,n}}\sqrt\pi
k_{n}(x)\approx\frac{(2n-1)!!}{x^{\,n+1}}.

Resposta

i_{n}(x)\approx\frac{x^{n}}{(2n+1)!!}, \qquad k_{n}(x)\approx\frac{(2n-1)!!}{x^{n+1}}, \quad x\ll1.

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