Modified Bessel function

The modified Bessel equation of order zero isIts solution regular at zero and normalized by $I_0(0)=1$ is the modified Bessel function $I_0$.

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 expansionsThe expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a differential equation. These formulas are documented in NIST DLMF, equations 10.40.1–2.

The solution of the Modified Bessel differential equation with power series $I_{\nu }(z)={\sum }_{j\ge 0}(z/2{)}^{2j+\nu }/[j!\Gamma (j+\nu +1)]$, interpreted through the reciprocal gamma function at its zeros. For fixed real order it grows exponentially along the positive real axis.

The differential equation defining modified Bessel functions of order $\nu$. For real fixed $\nu$ and positive $z$, its independent standard solutions are $I_{\nu }(z)$ and $K_{\nu }(z)$.

The modified Bessel function of the second kind is the solution $K_{\nu }(x)$ of the modified Bessel differential equation that decays exponentially as $x\to +\infty$.

For real $x>0$ and $\nu$, one useful representation isIt turns large-order asymptotics into a Laplace integral with a moving maximum.

For fixed $x>0$,as $\nu \to +\infty$. Applying Laplace's method to the integral representation of the modified Bessel function of the second kind proves the formula.

Laplace's integral method seeks $y(z)={\int }_C{e}^{zt}f(t)dt$. Substitution into the differential equation and integration by parts transfer multiplication by $z$ into differentiation with respect to $t$, producing a first-order equation for $f$ and endpoint conditions for $C$.

For $x>0$, substituting $y(x)={\int }_{\gamma }{e}^{xt}f(t)dt$ into $x{y}^{(3)}+2y=0$ and integrating by parts givesThe amplitude equation $(t^{3}f{)}^{\prime }=2f$ has solution $f(t)=C{t}^{-3}{e}^{-1/t^2}$. On $\gamma =(-\infty ,0)$ both endpoint terms vanish, and hence

For the Laguerre equationthe ansatz $y={\int }_{\gamma }{e}^{zt}f(t)dt$ givesThe contour must make the endpoint contributionvanish.

A finite contour based at an integrable branch point, or a closed contour around an integer-order pole, gives a solution analytic at $z=0$. It is therefore a constant multiple of the unique Frobenius power-series solution there.

For $\lambda =N\ge 0$, the amplitude $(t-1{)}^N/t^{N+1}$ has a pole at zero whose contour residue is a degree-$N$ Laguerre polynomial in $z$. For $\lambda =-m<0$, the amplitude $t^{m-1}/(t-1{)}^m$ has a pole at one whose residue is $e^z$ times a polynomial of degree $m-1$.

The normalized order-zero function has the representation