Bessel-Je Function Calculator

The calculator on this page calculates the logarithmically scaled modified Bessel function Jv(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.

By changing the 'Stretch Graphic' value, the X-axis scale can be lengthened or shortened.

Bessel-Je function Jv(x) of the first kind

The Bessel-Je function calculates the exponentially scaled Bessel function of the first kind. The Bessel-J functions exhibit oscillating (not exponential) behavior.

The result is a complex number if z is negative and n is a decimal number.

Plot of the Besselj function with atomic 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:

\(\displaystyle K_{\nu}(z) = \frac{\pi}{2} \frac{I_{-\nu}(z) - I_{\nu}(z)}{\sin(\nu \pi)} \)