Bessel function of the first kind
GPTKB entity
Statements (33)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:software
|
| gptkbp:application |
gptkb:signal_processing
heat conduction electromagnetic waves wave propagation vibrations of circular membranes |
| gptkbp:asymptoticForm |
J_n(x) ~ sqrt(2/πx) cos(x - nπ/2 - π/4) for large x
|
| gptkbp:category |
orthogonal functions
cylindrical functions |
| gptkbp:differential |
x^2 y'' + x y' + (x^2 - n^2) y = 0
|
| gptkbp:domain |
complex numbers
real numbers |
| gptkbp:field |
gptkb:mathematics
mathematical physics |
| gptkbp:firstZero |
positive real number
|
| gptkbp:hasIntegralRepresentation |
J_n(x) = (1/π) ∫_0^π cos(nτ - x sin τ) dτ
|
| https://www.w3.org/2000/01/rdf-schema#label |
Bessel function of the first kind
|
| gptkbp:namedAfter |
gptkb:Friedrich_Bessel
|
| gptkbp:order |
n
|
| gptkbp:orthogonal |
yes
|
| gptkbp:par |
J_{-n}(x) = (-1)^n J_n(x)
|
| gptkbp:recurrence |
J_{n-1}(x) - J_{n+1}(x) = 2J'_n(x)
|
| gptkbp:relatedTo |
gptkb:Bessel_function_of_the_second_kind
gptkb:Hankel_function gptkb:Modified_Bessel_function |
| gptkbp:seriesExpansion |
J_n(x) = (x/2)^n / Γ(n+1) Σ_{k=0}^∞ [(-1)^k (x^2/4)^k] / [k! Γ(n+k+1)]
|
| gptkbp:solvedBy |
Bessel's differential equation
|
| gptkbp:symbol |
J_n(x)
|
| gptkbp:usedIn |
gptkb:Fourier-Bessel_series
solutions to Helmholtz equation in cylindrical coordinates solutions to Laplace's equation in cylindrical coordinates |
| gptkbp:bfsParent |
gptkb:Bessel_functions
|
| gptkbp:bfsLayer |
6
|