log10I1

log10I1 denotes the base-10 logarithm of I1, most commonly the modified Bessel function of the first kind of order 1, evaluated at a real argument x. In conventional notation, log10I1(x) = log10(I1(x)). The expression is used when a logarithmic scale is applied to the magnitude of the Bessel function.

For real arguments, I1(x) is positive for x>0, with I1(0)=0 and I1(-x) = -I1(x). Therefore, log10I1(x) is real-valued only for x>0 (negative x yields a complex value if taken naively).

Asymptotic behavior helps intuition. Near the origin, I1(x) ~ x/2, so log10I1(x) ~ log10 x − log10 2. For large x, I1(x) ~ e^x / sqrt(2πx), yielding log10I1(x) ~ x/ln 10 − (1/2) log10(2πx). These approximations guide quick estimates and check numerical results.

Computational considerations: to compute log10I1(x) efficiently and without overflow, one may compute ln I1(x) and divide by ln 10, or use specialized routines for log of the Bessel function if available. For small or moderate x, direct evaluation of I1(x) followed by log10 is acceptable when numerical stability allows.

See also: Bessel I, log10, logarithms, special functions.