Calculate Hankel Function

Online calculator for Hankel functions H⁽¹⁾ᵥ(z) and H⁽²⁾ᵥ(z) - Cylindrical waves for incoming and outgoing wave propagation

Hankel Function Calculator

Hankel Functions (Cylindrical Waves)

The H⁽¹⁾ᵥ(z) and H⁽²⁾ᵥ(z) describe cylindrical waves with incoming and outgoing wave propagation.

Order of the Hankel function
Re(z) - Real part
Im(z) - Imaginary part
Result
H⁽¹⁾ᵥ(z):
H⁽²⁾ᵥ(z):

Hankel Function Curve

Mouse pointer on the graph shows the values.
The curve is only displayed for purely real arguments (Im(z) = 0).
Y-scale is limited to ±4 to resolve small values at z ≈ 0.

Incoming and Outgoing Cylindrical Waves

The Hankel functions describe the fundamental wave types in cylindrical geometries:

  • H⁽¹⁾ᵥ(z): Outgoing waves
  • H⁽²⁾ᵥ(z): Incoming waves
  • Complex conjugation: H⁽²⁾ᵥ(z) = [H⁽¹⁾ᵥ(z*)]* for real z
  • Asymptotic behavior: ~ √(2/πz) e^(±i(z-νπ/2-π/4))
  • Wave propagation: Describes far-field characteristics
  • Radiation condition: Physically correct boundary conditions

Applications of Hankel Functions

Hankel functions are essential for cylindrical wave problems and radiation theory:

Radiation Problems
  • Cylindrical antennas and waveguides
  • Electromagnetic scattering from cylinders
  • Acoustic radiation from rotationally symmetric sources
Wave Propagation
  • Green's functions for cylindrical geometries
  • Far-field approximations and asymptotic expansions
  • Boundary integral equations for exterior problems

Formulas for Hankel Functions

Definition via Bessel Functions
\[H_\nu^{(1)}(z) = J_\nu(z) + i Y_\nu(z)\] \[H_\nu^{(2)}(z) = J_\nu(z) - i Y_\nu(z)\]

Hankel functions of the first and second kind

Inverse Relations
\[J_\nu(z) = \frac{1}{2}[H_\nu^{(1)}(z) + H_\nu^{(2)}(z)]\] \[Y_\nu(z) = \frac{1}{2i}[H_\nu^{(1)}(z) - H_\nu^{(2)}(z)]\]

Representation of Bessel functions via Hankel functions

Asymptotic Forms (large z)
\[H_\nu^{(1)}(z) \sim \sqrt{\frac{2}{\pi z}} e^{i(z - \nu\pi/2 - \pi/4)}\] \[H_\nu^{(2)}(z) \sim \sqrt{\frac{2}{\pi z}} e^{-i(z - \nu\pi/2 - \pi/4)}\]

Outgoing and incoming cylindrical waves

Recurrence Formulas
\[\frac{2\nu}{z} H_\nu^{(k)}(z) = H_{\nu-1}^{(k)}(z) + H_{\nu+1}^{(k)}(z)\] \[\frac{d}{dz} H_\nu^{(k)}(z) = \frac{1}{2}[H_{\nu-1}^{(k)}(z) - H_{\nu+1}^{(k)}(z)]\]

For k = 1, 2 (same recurrences as Bessel functions)

Wronskian Determinants
\[W[H_\nu^{(1)}, H_\nu^{(2)}] = -\frac{4i}{\pi z}\] \[W[J_\nu, H_\nu^{(1)}] = \frac{2i}{\pi z}\]

Proves linear independence of function pairs

Symmetry Properties
\[H_{-n}^{(1)}(z) = (-1)^n H_n^{(2)}(z)\] \[H_{-n}^{(2)}(z) = (-1)^n H_n^{(1)}(z)\]

For integer n

Behavior as z → 0
\[H_\nu^{(1,2)}(z) \sim \pm \frac{2i}{\pi} \frac{\Gamma(\nu)}{\left(\frac{z}{2}\right)^\nu}\]

Singularity at origin for ν > 0

Special Values

Important Values
H₀⁽¹⁾(1) ≈ 0.765 + 0.088i H₁⁽¹⁾(1) ≈ 0.440 - 0.781i H₀⁽²⁾(1) ≈ 0.765 - 0.088i
Symmetry Properties
H⁽²⁾ᵥ(z*) = [H⁽¹⁾ᵥ(z)]*

For real z (complex conjugation)

Wave Character
\[|H_\nu^{(1,2)}(z)| \sim \sqrt{\frac{2}{\pi z}}\]

Amplitude decrease for large z

Complex Arguments
H⁽¹⁾ᵥ(z) and H⁽²⁾ᵥ(z)

Always yield complex numbers

Application Areas

Radiation problems, cylindrical waveguides, scattering theory, Green's functions.

Detailed Description of Hankel Functions

Mathematical Definition

The Hankel functions H⁽¹⁾ᵥ(z) and H⁽²⁾ᵥ(z) are complex linear combinations of the Bessel functions Jᵥ(z) and Yᵥ(z). They describe incoming and outgoing cylindrical waves and are fundamental for describing wave propagation problems.

Fundamental Property: H⁽¹⁾ᵥ(z) ↔ outgoing waves, H⁽²⁾ᵥ(z) ↔ incoming waves
Using the Calculator

Enter the order ν and the complex argument z = Re(z) + i Im(z). The calculator computes both Hankel functions simultaneously as complex numbers.

Physical Background

Hankel functions were named after the German mathematician Hermann Hankel (1839-1873). They are essential for describing wave propagation in cylindrical coordinates and enable the physically correct implementation of radiation conditions.

Properties and Applications

Physical Applications
  • Electromagnetic scattering from cylindrical objects
  • Acoustic radiation and wave propagation
  • Cylindrical antennas and waveguide theory
  • Green's functions for cylindrical geometries
Mathematical Properties
  • Complex functions with defined wave character
  • Asymptotic behavior: ~ √(2/πz) e^(±iz)
  • Linear independence among themselves and from Bessel functions
  • Satisfy Sommerfeld radiation condition
Numerical Aspects
  • Complex arithmetic: All results are complex numbers
  • Stability: Numerically stable for |z| > 1
  • Singularity: Problematic as z → 0 for ν > 0
  • Efficiency: Computed via Bessel-J and Y functions
Interesting Facts
  • H⁽¹⁾₀(z) describes the 2D Green's function of the Helmholtz equation
  • Hankel functions automatically satisfy the radiation condition
  • They are the cylindrical analogs to spherical Hankel functions
  • Essential for boundary integral equation methods

Calculation Examples and Wave Character

Small Argument

z = 0.5:

H⁽¹⁾₀(0.5) ≈ -0.177 + 0.512i

H⁽²⁾₀(0.5) ≈ -0.177 - 0.512i

Medium Argument

z = 2:

H⁽¹⁾₀(2) ≈ 0.224 + 0.510i

H⁽²⁾₀(2) ≈ 0.224 - 0.510i

Large Argument

z = 10:

|H⁽¹⁾₀(10)| ≈ 0.252

Asymptotic behavior

Detailed Physical Applications

Cylindrical Scattering

Scattered field:

ψ_scat = Σ aₙ H⁽¹⁾ₙ(kr) e^(inφ)

Outgoing waves from scatterer

Example: Electromagnetic scattering from conducting cylinders.

Green's Functions

2D Helmholtz equation:

G(r,r') = (i/4) H⁽¹⁾₀(k|r-r'|)

Fundamental solution for exterior problems

Application: Boundary integral equations for wave propagation.

Mathematical Properties and Relations

Asymptotic Behavior

For large z:

H⁽¹⁾ᵥ(z) ~ √(2/πz) e^(i(z-νπ/2-π/4))

H⁽²⁾ᵥ(z) ~ √(2/πz) e^(-i(z-νπ/2-π/4))

Shows the wave character clearly

Phase relationship: 180° phase difference between H⁽¹⁾ and H⁽²⁾.

Relations to Other Functions

Wronskian determinant:

W[H⁽¹⁾ᵥ, H⁽²⁾ᵥ] = -4i/(πz)

Relation to modified Bessel functions:

H⁽¹⁾ᵥ(iz) = (2i/π) e^(iνπ/2) Kᵥ(z)

Significance: Fundamental system for wave problems.

Special Orders and Their Physical Significance

Order ν = 0

H⁽¹⁾₀(z) - Fundamental solution:

\[H_0^{(1)}(z) = J_0(z) + i Y_0(z)\]

Green's function of the 2D Helmholtz equation

Application: Point sources in 2D wave propagation.

Order ν = 1

H⁽¹⁾₁(z) - Dipole characteristics:

\[H_1^{(1)}(z) = J_1(z) + i Y_1(z)\]

Important for dipole and gradient problems

Application: Cylindrical dipole antennas.

Numerical Computation and Algorithms

Computation Methods
  • Via Bessel functions: H⁽¹⁾ = J + iY, H⁽²⁾ = J - iY
  • Series expansion: For small z (complex arithmetic)
  • Asymptotic expansion: For large z (wave form)
  • Recurrence relations: For adjacent orders
Software Implementations
  • GNU GSL: Complex Hankel functions
  • Boost Math: C++ template library
  • SciPy: Python scipy.special.hankel1, hankel2
  • MATLAB: Built-in besselh function

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