I0lambda

I0lambda refers to the value of the modified Bessel function of the first kind of order zero, evaluated at a real argument λ. In many mathematical and applied contexts this function is written as I0(λ) or I_0(λ). The notation I0lambda is sometimes used as a shorthand or variable name in formulas and code.

I0(λ) is a classical special function with several equivalent representations. It has a power series expansion I0(λ) = sum from k=0 to ∞ of (λ^2/4)^k / (k!)^2. It also admits the integral representation I0(λ) = (1/π) ∫_0^π e^{λ cos θ} dθ, and, equivalently, I0(λ) = (1/2π) ∫_{-π}^{π} e^{λ cos θ} dθ. It is an even function, real-valued for real λ, and satisfies I0(0) = 1 and I0(λ) > 0 for all real λ.

I0(λ) is the solution of the modified Bessel differential equation x^2 y'' + x y' − x^2 y = 0. Its derivative is given by d/dλ I0(λ) = I1(λ), where I1 is the modified Bessel function of the first kind of order one. Asymptotically, for large |λ|, I0(λ) ~ e^{|λ|} / sqrt(2π|λ|). For small λ, I0(λ) ≈ 1 + (λ^2)/4 + (λ^4)/64 + ...

I0(λ) appears in various applications, including the Rice distribution in statistics, the 2D heat kernel and diffusion problems, and circular statistics where ratios of Bessel functions such as I1/I0 describe concentration measures. It is widely implemented in mathematical software libraries and is a standard special function in analysis and applied disciplines.

See also I1, Bessel functions of the first kind, and related modified Bessel functions.