Exponentially Scaled Bessel-Je Function for Complex Numbers

Calculation of the exponentially scaled Bessel function \(J_e(z) = e^{-|z|} J_\nu(z)\) of the first kind

Bessel-Je Function Calculator

Exponentially Scaled Bessel Function \(J_e(z)\)

The exponentially scaled Bessel function \(J_e(z) = e^{-|z|} J_\nu(z)\) prevents numerical overflows for large arguments and combines oscillatory behavior with numerical stability.

Complex Argument z = a + bi
+
i
Integer or rational order of the Bessel function
Calculation Result
\(J_e(z)\) =

Bessel-Je Properties

Scaling

Exponential

Factor: \(e^{-|z|}\)

Prevents overflow
Behavior

Oscillatory

Type: \(J_\nu(z)\)

Damped oscillation
Order

ν ∈ ℝ

Any real number

Integer or rational
Argument

z ∈ ℂ

Complex: a+bi

Real and imaginary part
Important Properties
  • Numerically stable calculation for large |z|
  • Combines oscillation with exponential damping
  • Defined as: \(J_e(z) = e^{-|z|} J_\nu(z)\)
  • Asymptotically: \(J_e(z) \sim \frac{1}{\sqrt{2\pi|z|}}\) for |z| → ∞
BesselJ

Plot of the Bessel-J function (before exponential scaling)

Definition of the Exponentially Scaled Bessel-Je Function

The exponentially scaled ordinary Bessel function \(J_e(z)\) is defined as:

Scaled Definition
\[J_e(z) = e^{-|z|} J_\nu(z) = e^{-|z|} \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m + \nu + 1)} \left(\frac{z}{2}\right)^{2m + \nu}\]

Exponentially scaled version to avoid numerical overflows in oscillating functions

Numerical Stability
\[|J_e(z)| \leq \frac{C}{\sqrt{|z|}}\]

Bounded for large |z|, prevents oscillation overflow

Relationship to Bessel-J
\[J_\nu(z) = e^{|z|} J_e(z)\]

Back-transformation to the original function

Important Properties of the Scaled Bessel-Je Function

Asymptotic Behavior
\[J_e(z) \sim \frac{1}{\sqrt{2\pi|z|}} \cos\left(|z| - \frac{\nu\pi}{2} - \frac{\pi}{4}\right) e^{-|z|}\]

Oscillation with exponential damping for large |z|

Numerical Advantages
\[\max|J_e(z)| < \infty \quad \text{for all } z\]

Bounded values prevent overflow despite oscillation

Scaling Factor
\[s(z) = e^{-|z|} = e^{-\sqrt{\text{Re}(z)^2 + \text{Im}(z)^2}}\]

Exponential damping factor based on the modulus of z

Recurrence Relations
\[J_{e,\nu-1}(z) + J_{e,\nu+1}(z) = \frac{2\nu}{z} J_{e,\nu}(z)\]

Scaled recurrence relations apply analogously

Applications of the Scaled Bessel-Je Function

Numerical Analysis
Large arguments Oscillation control Stable algorithms Precise calculation
Wave Physics
Cylindrical Waves:
Far-field analysis
Scattering problems
Acoustics:
Vibration modes
Resonance analysis
Electromagnetism

Waveguides

Antenna theory

Far-field patterns

Quantum Physics

Scattering theory

Cylindrical potentials

Asymptotic solutions

Phase shifts

Exponentially Scaled Bessel-Je Functions - Detailed Description

Numerical Stability with Oscillation

The exponentially scaled Bessel-Je function \(J_e(z)\) was developed to solve the numerical problems of the ordinary Bessel function \(J_\nu(z)\) for large arguments, where the oscillations can become very large.

Problem Statement:
• \(J_\nu(z)\) oscillates with increasing amplitude for large |z|
• Numerical instabilities in oscillating regions
• Loss of precision with large amplitudes
• Difficult calculation of differences

Solution Approach

By the definition \(J_e(z) = e^{-|z|} J_\nu(z)\), the oscillating function is provided with exponential damping, so that the resulting function remains numerically stable.

Advantages of Scaling

Without scaling: \(J_\nu(100)\) can show very large oscillations
With scaling: \(J_{e,\nu}(100)\) remains in the stable range

Mathematical Properties

The scaled function retains all important mathematical properties of the original oscillating Bessel function, but is numerically more stable.

Preserved Properties:
• Recurrence relations remain valid
• Oscillatory character preserved
• Symmetry relations persist
• Zeros are preserved

Implementation

In numerical practice, the scaled version is used for large arguments and the result is back-transformed if needed:

\[J_\nu(z) = e^{|z|} \cdot J_e(z)\]

Back-transformation only when explicitly needed

Computer Implementation

Modern libraries use the scaled version automatically for large arguments and handle the scaling transparently.

Comparison: Scaled vs. Unscaled

Unscaled Bessel-J Function
Definition: \(J_\nu(z)\)
Behavior: Oscillation with varying amplitude
Problems: Large amplitudes for large |z|
Numerics: Can become unstable
Scaled Bessel-Je Function
Definition: \(J_e(z) = e^{-|z|} J_\nu(z)\)
Behavior: Damped oscillation
Advantages: Stable amplitudes
Range: All |z| (practically unlimited)
Practical Application Guidelines
  • Small |z| ≤ 10: Both versions usable
  • Medium |z| ≤ 50: Scaled version recommended
  • Large |z| > 50: Only use scaled version
  • Oscillation analysis: Always scaled version
  • Far-field calculations: Prefer scaled version
  • Waveguide problems: Transparent scaling

Bessel Functions - Complete Definitions and Scaling

Ordinary Bessel Functions

The Bessel function of the first kind of order n is defined as:

\[J_{\nu}(z) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m + \nu + 1)} \left(\frac{z}{2}\right)^{2m + \nu}\]

The exponentially scaled version is:

\[J_e(z) = e^{-|z|} J_{\nu}(z)\]

The Bessel function of the second kind is:

\[Y_{\nu}(z) = \frac{J_{\nu}(z) \cos(\nu \pi) - J_{-\nu}(z)}{\sin(\nu \pi)}\]

Modified Bessel Functions

The modified Bessel function of the first kind is:

\[I_{\nu}(z) = i^{-\nu} J_{\nu}(iz)\]

The modified Bessel function of the second kind is:

\[K_{\nu}(z) = \frac{\pi}{2} \frac{I_{-\nu}(z) - I_{\nu}(z)}{\sin(\nu \pi)}\]
Application Areas

The exponentially scaled Bessel-Je function is particularly important for far-field analyses, wave propagation in cylindrical systems, scattering problems, and all areas where oscillating functions with large arguments occur. It enables stable calculations even with extreme oscillations.


More complex functions

Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •
Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •