modified Bessel function of the first kind

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:software
gptkbp:alsoKnownAs	gptkb:I-Bessel_function
gptkbp:appearsIn	solutions to Laplace's equation in cylindrical coordinates solutions to diffusion equations
gptkbp:application	gptkb:probability_theory gptkb:signal_processing engineering heat conduction mathematical physics wave propagation
gptkbp:asymptoticBehavior	I_n(x) ~ \\frac{e^x}{\\sqrt{2\\pi x}} as x \\to \\infty
gptkbp:domain	complex numbers
https://www.w3.org/2000/01/rdf-schema#label	modified Bessel function of the first kind
gptkbp:integration	I_n(x) = \\frac{1}{\\pi} \\int_0^\\pi e^{x \\cos \\theta} \\cos(n\\theta) d\\theta
gptkbp:isEntireFunction	true
gptkbp:namedAfter	gptkb:Friedrich_Bessel
gptkbp:order	n
gptkbp:orthogonality	not orthogonal
gptkbp:parameter	x
gptkbp:realArgument	x > 0
gptkbp:recurrence	I_{n-1}(x) - I_{n+1}(x) = \\frac{2n}{x} I_n(x)
gptkbp:relatedTo	gptkb:Bessel_function_of_the_first_kind
gptkbp:satisfies	gptkb:modified_Bessel_differential_equation
gptkbp:seriesExpansion	sum_{k=0}^\\infty \\frac{1}{k!\\,\\Gamma(n+k+1)}\\left(\\frac{x}{2}\\right)^{2k+n}
gptkbp:solvedBy	x^2 y'' + x y' - (x^2 + n^2) y = 0
gptkbp:symbol	I_n(x)
gptkbp:bfsParent	gptkb:Bessel_K
gptkbp:bfsLayer	7