Quick answer

Modified Bessel functions solve x2 y'' + x y′ − (x2 + v2) y = 0. Iv(x) grows exponentially for large real x; Kv(x) decays. They arise after imaginary scaling of cylindrical problems.

Formula

  • Modified equation: x2 y'' + x y′ − (x2 + v2) y = 0
  • Iv(x) = i−v Jv(ix), Kv related via analytic continuation
  • Large x: Iv(x) ~ e^x / √(2πx), Kv(x) ~ e−x √(π/(2x))

Introduction

Change the sign of the x2 term in Bessel's equation and oscillation turns into growth and decay. That is the modified family, central to heat diffusion, evanescent modes, and potential theory in cylinders. The Bessel Function Calculator evaluates cylindrical Jv, which you can relate to Iv through analytic continuation.

If cylindrical Jv is new, read first kind Jv first. If you need the singular partner on the cylindrical side, see Yv; on the modified side, Kv plays the analogous decaying role.

The home calculator targets Jv directly. Use complex mode with argument ix to explore how Iv(x) connects to Jv when you study the identity Iv(x) = i−v Jv(ix).

Steady heat in a long cylinder with fixed boundary temperature often involves I0 or K0 depending on whether the axis or infinity is included in the domain.

Do not assume Iv and Jv are interchangeable because their plots look similar near x = 1. Their large-x behavior diverges completely: one grows, one oscillates.

What are modified Bessel functions?

Iv(x) is the solution of the modified equation that stays finite at the origin, growing for large positive x. Kv(x) is the independent solution that decays for large positive x and blows up at zero.

In heat problems, radial modes that decay in time often involve K0(κ r) in a semi-infinite domain. In contrast, J0 describes standing waves, not exponential skin decay.

For complex arguments, branch choices matter. Stick to real x > 0 for first intuition.

The modified Wronskian Iv Kv' − Iv' Kv = 1/x parallels the cylindrical Wronskian for Jv and Yv and supports the same style of numerical checks.

Bessel functions of the third kind (Kelvin functions) appear in vibrating plates; they are further relatives you meet only in specialized engineering tables.

Key relations

  • I−v(x) = Iv(x) for many standard definitions
  • Wronskian: Iv Kv' − Iv' Kv = 1/x
  • Connection: Iv(x) = i−v Jv(ix)

The connection formula explains how to check Iv at a real x using complex Jv with argument ix, when your tool supports complex mode.

Recurrence relations parallel the Jn family with sign changes from the modified equation.

For large x, plotting log |Iv| versus x should be nearly linear with slope 1 before subdominant terms matter. Kv shows the opposite slope.

Study modified functions alongside Jv

  1. Identify the equation sign If your model has −x2 instead of +x2 on y, you need Iv and Kv, not Jv and Yv.
  2. Pick Iv or Kv by boundary behavior Finite at origin → Iv. Decay at infinity on (0,∞) → Kv. Annular regions may mix both.
  3. Check via complex Jv when possible Use complex mode with argument ix to explore Iv(x) numerically if you lack a dedicated Iv button.
  4. Plot exponential scales carefully Use log scale for large x when comparing Iv and Kv in the same figure.

Example: I0(1) versus J0(1)

I0(1) ≈ 1.2661, noticeably larger than J0(1) ≈ 0.7652 because the modified function accumulates exponential growth for larger arguments.

K0(1) ≈ 0.4210 is finite away from zero but logarithmically singular as x → 0+, similar in role to Y0 though on a different equation.

At x = 2, I0(2) ≈ 2.2796 while J0(2) ≈ 0.2239, showing how quickly the two families separate once x leaves the neighborhood of the origin.