Evaluate first-, second-, and third-kind Bessel functions for a real order and a real or complex argument.
Mathematical Curiosities & Special Topics
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Try an example
Choose a real or complex argument, then enter the real order and labelled argument components. Complex results use the principal branch. Inputs stay on this device.
Result
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A mathematical curiosity makes a surprising pattern, process, game, approximation, or paradox explicit enough to explore step by step.
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The relationship
Write the rule before calculating
Jᵥ(z)=Σₖ₌₀∞(−1)^k(z/2)^(2k+ᵥ)/[k!Γ(k+ᵥ+1)]; Yᵥ and Hᵥ⁽¹,²⁾ follow from Jᵥ and J₋ᵥ.
See the structure
What the calculation is doing
startapply rulenext staterepeatobserve patternJᵥ(z)=Σₖ₌₀∞(−1)^k(z/2)^(2k+ᵥ)/[k!Γ(k+ᵥ+1)]; Yᵥ and Hᵥ⁽¹,²⁾ follow from Jᵥ and J₋ᵥ.
Worked interpretation
Read the result in context
For ν=0 and z=1: J₀≈0.7651977, Y₀≈0.0882570, and H₀⁽¹⁾=J₀+iY₀.
Interpret with care
Important boundary
Explorers use the stated finite inputs, simulation limits, and conventions; a result should not be generalized beyond that model.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Quick guide
How to use this calculator
Choose whether the argument is real or complex; complex mode reveals a separate imaginary-component box.
Enter a real order ν and the real and, when applicable, imaginary components of z.
Compare Jν, Yν, and the two Hankel functions, then use magnitude, phase, and the real-axis plot to interpret the result.
Calculation method
Calculate the complete Bessel-function family
Jᵥ(z)=Σₖ₌₀∞(−1)^k(z/2)^(2k+ᵥ)/[k!Γ(k+ᵥ+1)]; Yᵥ and Hᵥ⁽¹,²⁾ follow from Jᵥ and J₋ᵥ.
The second-kind value is constructed from Jν and J−ν, with a symmetric limiting calculation at integer orders. The Hankel functions combine Jν and Yν as Hν⁽¹⁾=Jν+iYν and Hν⁽²⁾=Jν−iYν.
Worked example
Worked example
For ν=0 and z=1: J₀≈0.7651977, Y₀≈0.0882570, and H₀⁽¹⁾=J₀+iY₀.
Jᵥ(z)=Σₖ₌₀∞(−1)^k(z/2)^(2k+ᵥ)/[k!Γ(k+ᵥ+1)]; Yᵥ and Hᵥ⁽¹,²⁾ follow from Jᵥ and J₋ᵥ.
Supported inputs
Precision and limits
Model scope
The order is real; the argument may be real or complex. Complex powers follow the principal branch, so branch conventions matter for non-integer orders.
Deterministic versus simulated
Jν and J−ν use convergent power series. Integer-order Yν is evaluated through the defining order limit; all values use finite double precision.
Numerical methods
Finite double-precision arithmetic is used for probability models and special functions. Gamma uses a bounded Lanczos approximation, Bessel functions use bounded convergent series and order limits, and erf/erfc disclose their approximation error.
Limits
Yν(z) and both Hankel functions are singular at z=0. Values near singularities or severe cancellation can lose precision and are rejected when the supported method cannot return a finite result.
Calculator-specific rule
The order ν is real from −20 to 20. Real and imaginary argument components are limited to −10 through 10. Complex powers use the principal branch; z=0 is singular for Y and Hankel functions.
