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.

Description

Kelvin ber is the real component of J0 evaluated on a rotated complex argument.

Kelvin ber is the real component of J0 evaluated on a rotated complex argument. The rotation J0(x exp(3πi/4)) fixes the sign convention.^{[1, 2]}

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.

When to use Kelvin Ber Approximation

Use this tool to 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.

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

How to use Kelvin Ber 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 kelvinBerValue. 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

Enter finite real x with |x|≤20. Accumulate the even-index terms of the rotated Bessel power series with recurrence, stopping at relative term criterion 10⁻¹⁶ within 200 iterations; nonconvergence is rejected.

ber(x) = real 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
kelvinBerValue	Kelvin ber value	number

json

{
  "input": {
    "x": 1
  },
  "output": {
    "kelvinBerValue": 0.984381781213087
  }
}

Limitations and assumptions

This bounded real approximation has no complex continuation or certified roundoff bound.

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

Similar or alternative tools

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.
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.