Modified Bessel Function Calculator Iₙ(x) and Kₙ(x)

Enter an order n and x to evaluate the modified Bessel functions of the first kind Iₙ(x) and second kind Kₙ(x), with neighboring orders, a graph and an order-by-order table.

Input

Enter an order n and x to compute the modified Bessel functions of the first kind Iₙ(x) and second kind Kₙ(x). Here x is a positive real number and n is a non-negative integer.

Order n

Non-negative integer

Value of x

Positive real number

Result

First kind I_0(1)

1.2660658778

Second kind K_0(1)

0.42102444

Previous order of I (next to n = 0)

0.565159104

Next order of I (n+1)

0.565159104

Next order of K (n+1)

0.60190723

Graph of Iₙ(x) and Kₙ(x) (n = 0, log vertical axis)

Iₙ(x) (first kind, n = 0)

Kₙ(x) (second kind, n = 0)

Values by order

Order n	First kind Iₙ(x)	Second kind Kₙ(x)
0	1.26606588	0.42102444
1	0.5651591	0.60190723
2	0.02216842	1.6248389
3	0.00273712	7.10126282

How it works

The modified Bessel functions solve the modified Bessel equation x^2 y'' + x y' - (x^2 + n^2) y = 0. The first kind Iₙ(x) grows for large x, while the second kind Kₙ(x) decays exponentially as x increases.
For small x a power series is used. The first kind is Iₙ(x) = Σ (x/2)^(2k+n) / (k! (k+n)!), and high orders are obtained stably by Miller backward recurrence, normalized with I₀(x).
For large x asymptotic expansions are used: I₀ and I₁ as e^x / sqrt(2 pi x) times a series, and Kₙ as sqrt(pi/(2x)) e^(-x) times a series.
The second kind starts from K₀ and K₁ via logarithmic series, then climbs orders with the upward recurrence K(n+1) = K(n-1) + (2n/x) Kₙ. The relation I(n+1) = I(n-1) - (2n/x) Iₙ is also used.
The input x must be a positive real number, and the order n must be a non-negative integer. Displayed values are approximations with practical accuracy.

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