Kelvin Bei Approximation

Evaluate Kelvin bei(x), the imaginary component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit.

Description

Kelvin bei is the imaginary component of a rotated Bessel function.

Kelvin bei is the imaginary component of a rotated Bessel function. Its convention here gives a positive leading x²/4 term.^{[1, 2]}

Evaluate Kelvin bei(x), the imaginary component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit.

When to use Kelvin Bei Approximation

Use this tool to evaluate Kelvin bei(x), the imaginary component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Kelvin Bei Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter x using the field table and calculation rules, then run the tool.
Read kelvinBeiValue. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
x	X	number	Required

Use finite real x with |x|≤20. Accumulate odd-index contributions from the rotated power series and stop when the shared term magnitude falls below 10⁻¹⁶ times the largest of 1 and component magnitudes, within 200 iterations.

bei(x) = imaginary part of J0(x exp(3πi/4))

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
kelvinBeiValue	Kelvin bei value	number

json

{
  "input": {
    "x": 1
  },
  "output": {
    "kelvinBeiValue": 0.24956604003665972
  }
}

Limitations and assumptions

This is a bounded floating approximation; failed series convergence is rejected.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Kelvin Ber Approximation: Evaluate Kelvin ber(x), the real component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit. For a complementary task, use Bessel J0 Approximation: Evaluate real Bessel J0 by bounded convergent power series on |x|≤20; outside-series regime refused rather than an unstated asymptotic formula.

Further background: Kelvin functions: definitions and basic properties.

References

Kelvin Bei Approximation — tool.io
Kelvin functions: definitions and basic properties — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Kelvin Ber Approximation
Evaluate Kelvin ber(x), the real component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit.
Bessel J0 Approximation
Evaluate real Bessel J0 by bounded convergent power series on |x|≤20; outside-series regime refused rather than an unstated asymptotic formula.