Calculate Spherical Bessel Function yv(z)
Online calculator for computing the spherical Bessel function of the second kind
Spherical Bessel-Y Function Calculator
Spherical Neumann Function
The yv(z) or spherical Neumann function shows singular behavior at the origin and is the second solution in spherical coordinates.
Spherical Bessel-Y Function Curve
Mouse pointer on the graph shows the values.
The spherical Neumann function shows singularities at the origin and large oscillations.
Properties of the Spherical Neumann Function
The spherical Neumann function is the second linearly independent solution:
- Singularity: yv(z) → -∞ for z → 0
- Relation to Yv: yv(z) = √(π/2z) Yv+1/2(z)
- Asymptotic behavior: Oscillation with 1/z damping
- Physical meaning: Outgoing waves
- Combination with jv: General solution possible
- Application: Scattering problems and boundary conditions
Spherical Neumann Functions in Physics
The spherical Neumann function forms a complete solution system with jv:
General Solution
Linear combination of both solutions
Hankel Functions
Incoming and outgoing waves
Formulas for the Spherical Neumann Function
Definition via Classical Neumann Function
Relation to the classical Neumann function Yν
Explicit Expressions for Small Orders
Simple trigonometric representation with singularity
Asymptotic Form
For large z (90° phase shift to jν)
Wronskian Determinant
Proof of linear independence
Recurrence Formulas
Identical to jν (same differential equation)
Special Properties
Singularity at Origin
For all ν ≥ 0
Important Values
Zeros of y₀
Zeros at odd multiples of π/2
Behavior for Large z
Oscillation with 1/z damping
Application Areas
Scattering problems, outgoing waves, boundary conditions for infinite domains.
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)
Spherical Neumann functions are analogous to spherical Bessel functions, but with a different definition. They also occur in spherical geometry. They are defined as:
\(\displaystyle y_\mu(z) = \sqrt{\frac{\pi}{2z}} Y_{\mu+1/2}(z) \)
where \(Y_\mu(z)\) is the Bessel function of the second kind (Neumann function). These functions show singular behavior at the origin and are important for scattering problems.
|
|