Calculate Spherical Bessel J Function jv(x)

Online calculator for computing the spherical Bessel function of the first kind

Spherical Bessel-J Function Calculator

Spherical Bessel Function

The jv(x) or spherical Bessel function is a special form of the classical Bessel function for spherical coordinates.

Order of the spherical Bessel function
Function argument (x > 0)
X-axis scaling
Result
jv(x):

Spherical Bessel-J Function Curve

Mouse pointer on the graph shows the values.
The spherical Bessel function shows characteristic oscillations in spherical coordinates.

Properties of the Spherical Bessel-J Function

The spherical Bessel function is a specialized form for spherical geometry:

  • Spherical coordinates: jv(x) for sphere problems
  • Relation to Jv: jv(x) = √(π/2x) Jv+1/2(x)
  • Asymptotic behavior: Oscillation with 1/x damping
  • Application: Electromagnetic waves in spheres
  • Quantum mechanics: Radial wave functions
  • Acoustics: Sound waves in spherical cavities

Spherical Bessel Functions in the Helmholtz Equation

The spherical Bessel-J function is the regular solution of the Helmholtz equation in spherical coordinates:

Helmholtz Equation (spherical)
\[\nabla^2 \psi + k^2 \psi = 0\]

Separated in spherical coordinates

Radial Solution
\[R_l(kr) = j_l(kr)\]

Regular solution at origin

Formulas for the Spherical Bessel-J Function

Definition via Classical Bessel Function
\[j_\nu(x) = \sqrt{\frac{\pi}{2x}} J_{\nu+1/2}(x)\]

Fundamental relation to the classical Bessel function

Explicit Expressions for Small Orders
\[j_0(x) = \frac{\sin x}{x}, \quad j_1(x) = \frac{\sin x}{x^2} - \frac{\cos x}{x}\]

Simple trigonometric representation

Asymptotic Form
\[j_\nu(x) \sim \frac{1}{x} \cos\left(x - \frac{(\nu+1)\pi}{2}\right)\]

For large x (simpler than classical Bessel function)

Recurrence Formulas
\[\frac{2\nu+1}{x} j_\nu(x) = j_{\nu-1}(x) + j_{\nu+1}(x)\] \[\frac{d}{dx} j_\nu(x) = \frac{\nu}{x} j_\nu(x) - j_{\nu+1}(x)\]

Recurrence relations for spherical Bessel functions

Orthogonality Relation
\[\int_0^R r^2 j_l(k_m r) j_l(k_n r) dr = \frac{R^3}{2} \delta_{mn} [j_{l+1}(k_m R)]^2\]

Orthogonality on spherical shells

Special Values

Important Values
j₀(0) = 1 j₁(0) = 0 j₀(π) = 0
Zeros of j₀
π, 2π, 3π, 4π, ...

Zeros at integer multiples of π

Behavior at x = 0
\[j_\nu(0) = \begin{cases} 1 & \text{if } \nu = 0 \\ 0 & \text{if } \nu > 0 \end{cases}\]

Regular behavior at origin

Application Areas

Quantum mechanics, electromagnetic scattering from spheres, acoustics in spherical cavities, atomic physics.

Description and Formulas

The spherical Bessel functions are a special class of functions that play an important role in physics and mathematics. They are solutions to the Bessel 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 the Bessel differential equation and are often called cylinder functions. The Bessel function of the first kind of order n is defined as:

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

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

Spherical Bessel Functions (jμ)

These functions are special Bessel functions that occur in spherical geometry. They are solutions to 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 used, for example, in describing electromagnetic waves in spherical coordinates.

Spherical Neumann Functions (yμ)

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

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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 Zeta