Spherical Bessel J function j_v(x) calculator

The calculator on this page calculates the spherical Bessel function j_v(x) 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 x. Then click the 'Calculate' button. The ordinal number must be an integer and the argument x must be positive.

Changing the Stretch Graph value can lengthen or shorten the X-axis scale.

Description and formulas

The spherical Bessel functions are a special class of functions, which play an important role in physics and mathematics. They are solutions of Bessel's differential equation, which represents the radial part of the Laplace equation with cylindrical or spherical symmetry.

Bessel functions of the first kind (Jν)

These functions are solutions to Bessel's differential equation and are often referred to as cylindrical functions. The Bessel function of the first genus of the nth order is defined as:

\(\displaystyle J_{\nu}(x) = \frac{(x/2)^{\nu}}{\Gamma(\nu + 1)} \, {}_0F_1(; \nu + 1; - x^2/4) \)

Where \(\Gamma(\nu + 1)\) is the gamma function and \(\nu\) is a real or complex number. These functions appear in various physical problems, such as the investigation of natural vibrations of a circular membrane, heat conduction in rods or field distribution in circular waveguides¹.

Spherical Bessel functions (jμ)

These functions are special Bessel functions that occur in spherical geometry. They are solutions of the Helmholtz equation in spherical coordinates. The spherical Bessel function jμ is defined as:

\(\displaystyle j_{\mu}(x) = \sqrt{\frac{\pi}{2x}} J_{\mu+1/2}(x) \)

Here \(\mu\) is an integer or half integer order. Spherical Bessel functions are, for example used in the description of electromagnetic waves in spherical coordinates.

Spherical Neumann functions (yμ)

These are analogous to the spherical Bessel functions, but with a different definition. They also occur in spherical geometry.

\(\displaystyle j_n(x)=\sqrt{\frac{π}{2x}}J_{n+\frac{1}{2}}(x)\)