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The modified Bessel function () = (/) / 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):
However, the modified Bessel functions of the second kind () also satisfy the same recurrence relation = + + (). The first solution decreases rapidly with . The second solution increases rapidly with . Miller's algorithm provides a numerically stable procedure to obtain the decreasing solution.
where K ν (z) is the ν th order modified Bessel function of the second kind. These functions are named after William Thomson, 1st Baron Kelvin. While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments xe iφ, 0 ≤ φ ...
The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its ... is a modified Bessel function of the second kind, y n (x) ...
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where = is the shape factor, = / is the scale factor, and is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting α = 1 {\displaystyle \alpha =1} , v = β {\displaystyle v=\beta } , and b = β / μ {\displaystyle b=\beta /\mu } , albeit with different physical interpretation of b ...
where K p is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen.
where is the modified Bessel function of the second kind. As t → ∞ {\displaystyle t\rightarrow \infty } , the solution approaches that of a rigid vortex. The force per unit area exerted on the cylinder is