Bessel-Ke Funktion für reelle Zahlen
Online calculator for calculating the log. modified Bessel function Kn(z) of the second kind
The calculator on this page calculates the logarithmically modified Bessel function Kn(z) for real numbers. Bessel functions for complex numbers can be found in the complex numbers section.
To calculate, enter the ordinal number and the argument z. Then click the 'Calculate' button. The ordinal number must be an integer and the argument z must be positive.
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Bessel-Ke function of the second kind
The Bessel-Ke function calculates the exponentially scaled modified Bessel function Kn(z) of the second kind. The modified Bessel functions do not exhibit oscillating but rather exponential behavior. BesselKe (n, z) is a solution of the modified Bessel differential equation.
The result can be a complex number if z is negative and n is a decimal number.
Plot of the Bessel-K function with ordinal numbers 0, 1 and 2
Definition of the Bessel functions
Bessel functions of the first kind (Jν)
The Bessel function of the first genus of the nth order is defined as:
\(\displaystyle J_{\nu}(z) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m + \nu + 1)} \left(\frac{z}{2}\right)^{2m + \nu} \)
Here \( \Gamma \) is the gamma function.
At the origin (\( z = 0 \)) these functions are finite for integer values of \( \nu \).
For non-integer values of \( \nu \) there are two linearly independent solutions.
For integer values of \( \nu \) the relationship holds:
\(\displaystyle J_{-\nu}(z) = (-1)^{\nu} J_{\nu}(z) \)
Bessel functions of the second genus (Yν)
The Bessel function of second genus nth order is defined as:
\(\displaystyle Y_{\nu}(z) = \frac{J_{\nu}(z) \cos(\nu \pi) - J_{-\nu}(z)}{\sin(\nu \ pi)} \)
Modified Bessel functions (Iν, Kν)
The modified Bessel function of the first kind of nth order is defined as:
\(\displaystyle I_{\nu}(z) = i^{-\nu} J_{\nu}(iz) \)
The modified Bessel function of the second type nth order is defined as:
\(\displaystyle K_{\nu}(z) = \frac{\pi}{2} \frac{I_{-\nu}(z) - I_{\nu}(z)}{\sin(\nu \pi)} \)
IT Functions
Decimal, Hex, Bin, Octal conversion • Shift bits left or right • Set a bit • Clear a bit • Bitwise AND • Bitwise OR • Bitwise exclusive ORSpecial functions
Airy • Derivative Airy • Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye • Spherical-Bessel-J • Spherical-Bessel-Y • Hankel • Beta • Incomplete Beta • Incomplete Inverse Beta • Binomial Coefficient • Binomial Coefficient Logarithm • Erf • Erfc • Erfi • Erfci • Fibonacci • Fibonacci Tabelle • Gamma • Inverse Gamma • Log Gamma • Digamma • Trigamma • Logit • Sigmoid • Derivative Sigmoid • Softsign • Derivative Softsign • Softmax • Struve • Struve table • Modified Struve • Modified Struve table • Riemann ZetaHyperbolic functions
ACosh • ACoth • ACsch • ASech • ASinh • ATanh • Cosh • Coth • Csch • Sech • Sinh • TanhTrigonometrische Funktionen
ACos • ACot • ACsc • ASec • ASin • ATan • Cos • Cot • Csc • Sec • Sin • Sinc • Tan • Degree to Radian • Radian to Degree
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