structural:expansion
Modified Bessel functions: I, K

The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as I α (x) = i − α J α (i x) = ∑ m = 0 ∞ 1 m ! Γ (m + α + 1) (x 2) 2 m + α, K α (x) = π 2 I − α (x) − I α (x) sin α π, {\displaystyle {\begin{aligned}I_{\alpha }(x)&=i^{-\alpha }J_{\alpha }(ix)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\alpha +1)}}\left({\frac {x}{2}}\right)^{2m+\alpha },\\K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}} when α is not an integer; when α is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x. The series expansion for I (x) is thus similar to that for J (x), but without the alternating (−1) factor.

K α {\displaystyle K_{\alpha }} can be expressed in terms of Hankel functions: K α (x) = { π 2 i α + 1 H α (1) (i x) − π < arg x ≤ π 2 π 2 (− i) α + 1 H α (2) (− i x) − π 2 < arg x ≤ π {\displaystyle K_{\alpha }(x)={\begin{cases}{\frac {\pi }{2}}i^{\alpha +1}H_{\alpha }^{(1)}(ix)&-\pi <\arg x\leq {\frac {\pi }{2}}\\{\frac {\pi }{2}}(-i)^{\alpha +1}H_{\alpha }^{(2)}(-ix)&-{\frac {\pi }{2}}<\arg x\leq \pi \end{cases}}}

Using these two formulae the result to J α 2 (z) {\displaystyle J_{\alpha }^{2}(z)} + Y α 2 (z) {\displaystyle Y_{\alpha }^{2}(z)}, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following J α 2 (x) + Y α 2 (x) = 8 π 2 ∫ 0 ∞ cosh (2 α t) K 0 (2 x sinh t) d t, {\displaystyle J_{\alpha }^{2}(x)+Y_{\alpha }^{2}(x)={\frac {8}{\pi ^{2}}}\int _{0}^{\infty }\cosh(2\alpha t)K_{0}(2x\sinh t)\,dt,}

given that the condition Re(x) > 0 is met. It can also be shown that J α 2 (x) + Y α 2 (x) = 8 cos (α π) π 2 ∫ 0 ∞ K 2 α (2 x sinh t) d t, {\displaystyle J_{\alpha }^{2}(x)+Y_{\alpha }^{2}(x)={\frac {8\cos(\alpha \pi)}{\pi ^{2}}}\int _{0}^{\infty }K_{2\alpha }(2x\sinh t)\,dt,}

only when | Re(α) | < 1 / 2 and Re(x) ≥ 0 but not when x = 0.

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if − π < arg z ≤ π / 2): J α (i z) = e α π i 2 I α (z), Y α (i z) = e (α + 1) π i 2 I α (z) − 2 π e − α π i 2 K α (z). {\displaystyle {\begin{aligned}J_{\alpha }(iz)&=e^{\frac {\alpha \pi i}{2}}I_{\alpha }(z),\\Y_{\alpha }(iz)&=e^{\frac {(\alpha +1)\pi i}{2}}I_{\alpha }(z)-{\tfrac {2}{\pi }}e^{-{\frac {\alpha \pi i}{2}}}K_{\alpha }(z).\end{aligned}}}

I (x) and K (x) are the two linearly independent solutions to the modified Bessel's equation: x 2 d 2 y d x 2 + x d y d x − (x 2 + α 2) y = 0. {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y=0.}

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, I and K are exponentially growing and decaying functions respectively. Like the ordinary Bessel function J, the function I goes to zero at x = 0 for α > 0 and is finite at x = 0 for α = 0. Analogously, K diverges at x = 0 with the singularity being of logarithmic type for K, and 1 / 2 Γ(| α |)(2/ x) otherwise.

Two integral formulas for the modified Bessel functions are (for Re(x) > 0): I α (x) = 1 π ∫ 0 π e x cos θ cos α θ d θ − sin α π π ∫ 0 ∞ e − x cosh t − α t d t, K α (x) = ∫ 0 ∞ e − x cosh t cosh α t d t. {\displaystyle {\begin{aligned}I_{\alpha }(x)&={\frac {1}{\pi }}\int _{0}^{\pi }e^{x\cos \theta }\cos \alpha \theta \,d\theta -{\frac {\sin \alpha \pi }{\pi }}\int _{0}^{\infty }e^{-x\cosh t-\alpha t}\,dt,\\K_{\alpha }(x)&=\int _{0}^{\infty }e^{-x\cosh t}\cosh \alpha t\,dt.\end{aligned}}}

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Re(ω) > 0): 2 K 0 (ω) = ∫ − ∞ ∞ e i ω t t 2 + 1 d t. {\displaystyle 2\,K_{0}(\omega)=\int _{-\infty }^{\infty }{\frac {e^{i\omega t}}{\sqrt {t^{2}+1}}}\,dt.}

It can be proven by showing equality to the above integral definition for K. This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions K and K can be represented in terms of rapidly convergent integrals K 1 3 (ξ) = 3 ∫ 0 ∞ exp (− ξ (1 + 4 x 2 3) 1 + x 2 3) d x, K 2 3 (ξ) = 1 3 ∫ 0 ∞ 3 + 2 x 2 1 + x 2 3 exp (− ξ (1 + 4 x 2 3) 1 + x 2 3) d x. {\displaystyle {\begin{aligned}K_{\frac {1}{3}}(\xi)&={\sqrt {3}}\int _{0}^{\infty }\exp \left(-\xi \left(1+{\frac {4x^{2}}{3}}\right){\sqrt {1+{\frac {x^{2}}{3}}}}\right)\,dx,\\K_{\frac {2}{3}}(\xi)&={\frac {1}{\sqrt {3}}}\int _{0}^{\infty }{\frac {3+2x^{2}}{\sqrt {1+{\frac {x^{2}}{3}}}}}\exp \left(-\xi \left(1+{\frac {4x^{2}}{3}}\right){\sqrt {1+{\frac {x^{2}}{3}}}}\right)\,dx.\end{aligned}}}

The modified Bessel function K 1 2 (ξ) = (2 ξ / π) − 1 / 2 exp (− ξ) {\displaystyle K_{\frac {1}{2}}(\xi)=(2\xi /\pi)^{-1/2}\exp(-\xi)} is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.

The modified Bessel function of the second kind has also been called by the following names (now rare):