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Scaling keeps the modified Bessel value bounded as the unscaled value grows

Statement

For $n = 0$ and $x \geq 0$ , the scaled value $e^{- x} I_{0} (x)$ lies in $[0, 1]$ , while $I_{0} (x) \to \infty$ as $x \to \infty$ . Thus any fixed finite output range for unscaled values is eventually exceeded, whereas the scaled value remains bounded and can be evaluated directly from its scaled integral.

Facts & Assumptions

Given: Real $x \geq 0$ .

[F1]

Integral representations and a range bound for integer Bessel functions: $I_{0} (x) = \frac{1}{π} \int_{0}^{π} e^{x \cos θ} d θ, 0 \leq e^{- x} I_{0} (x) \leq 1.$

[F2]

The derivatives of sine and cosine are cosine and minus sine: $\sin^{'} t = \cos t$ , $\cos^{'} t = - \sin t$ , $\sin 0 = 0$ , and $\cos 0 = 1$ .

[F3]

Signs, monotonicity intervals, and ranges of sine and cosine: $\cos t \leq 1$ for all real $t$ .

[F4]

The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: the continuous function $\sin t$ is integrable, and its primitive $- \cos t$ gives $\int_{0}^{θ} \sin t d t = 1 - \cos θ$ .

Proof

1.1F2F3F4algebra

Let $h (t) = t - \sin t$ . By [F2], $h (0) = 0$ and $h^{'} (t) = 1 - \cos t \geq 0$ for $t \geq 0$ by [F3], so $\sin t \leq t$ . Again by [F2] and [F4], $1 - \cos θ = \int_{0}^{θ} \sin t d t \leq \int_{0}^{θ} t d t = θ^{2} / 2 (0 \leq θ \leq 1) .$

2.1F1step 1.1algebra

For $x \geq 1$ and $0 \leq θ \leq 1 / \sqrt{x}$ , step 1.1 gives $x \cos θ \geq x - 1 / 2$ . Restricting the integral in [F1] to this interval therefore yields $I_{0} (x) \geq \frac{e^{x - 1 / 2}}{π \sqrt{x}} .$ The exponential series gives $e^{x} \geq x^{2} / 2$ for $x \geq 0$ , so this lower bound is at least $e^{- 1 / 2} x^{3 / 2} / (2 π)$ and tends to infinity.

3.1F1step 2.1∎

The scaled range in [F1] holds for every $x \geq 0$ , while step 2.1 shows the unscaled value exceeds any fixed finite range for sufficiently large $x$ . The scaled integral method of Composite midpoint enclosure for the scaled modified Bessel integral evaluates the bounded scaled quantity directly.

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Sources

DLMF, §10.40, asymptotic expansions for modified Bessel functions (standard reference, not scraped)
DLMF, §§10.25 and 10.40 (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §2 (standard reference, not scraped)