Calculate Struve Function

Online calculator and formulas for computing the Struve function Hv(x)

Struve Function Calculator

Inhomogeneous Bessel Equation

The Hv(x) or Struve function is the particular solution of the inhomogeneous Bessel differential equation.

Order of the Struve function (0 or 1)
Function argument (x ≥ 0)
Result
Hv(x):

Struve Function Curve

Mouse pointer on the graph shows the values.
The Struve function shows characteristic properties of the inhomogeneous Bessel equation.

Properties of the Struve Function

The Struve function is the particular solution of the inhomogeneous Bessel differential equation:

  • Inhomogeneous ODE: x²y'' + xy' + (x² - v²)y = f(x)
  • Special inhomogeneity: f(x) = 4(x/2)^(v+1)/[√π Γ(v+1/2)]
  • Asymptotic behavior: Hv(x) → Yv(x) + const.
  • Physical meaning: Forced oscillations
  • Relation to Bessel: Supplements homogeneous solutions
  • Application: Wave packets and excitation problems

Inhomogeneous Bessel Equation and Its Solutions

The Struve function extends the solution spectrum of the Bessel equation:

Inhomogeneous Bessel ODE
\[x^2y'' + xy' + (x^2 - v^2)y = f(x)\]

With special inhomogeneity f(x)

General Solution
\[y(x) = AJ_v(x) + BY_v(x) + H_v(x)\]

Homogeneous + particular solution

Formulas for the Struve Function

Series Expansion
\[H_v(x) = \left(\frac{x}{2}\right)^{v+1} \sum_{k=0}^{\infty} \frac{(-1)^k \left(\frac{x}{2}\right)^{2k}}{\Gamma\left(k+\frac{3}{2}\right)\Gamma\left(k+v+\frac{3}{2}\right)}\]

Fundamental power series expansion

Asymptotic Form
\[H_v(x) \sim Y_v(x) + \frac{1}{\pi} \sum_{k=0}^{n-1} \frac{\Gamma(k-v+\frac{1}{2})}{\Gamma(\frac{1}{2})} \left(\frac{x}{2}\right)^{v-2k-1}\]

For large x (relation to Neumann function)

Recurrence Formulas
\[\frac{2v}{x} H_v(x) = H_{v-1}(x) + H_{v+1}(x) + \frac{2}{\sqrt{\pi}\Gamma(v+\frac{1}{2})} \left(\frac{x}{2}\right)^v\]

Modified recurrence with inhomogeneity term

Integral Representation
\[H_v(x) = \frac{2\left(\frac{x}{2}\right)^v}{\sqrt{\pi}\Gamma\left(v+\frac{1}{2}\right)} \int_0^1 (1-t^2)^{v-\frac{1}{2}} \sin(xt) dt\]

Integral form for v > -1/2

Relation to Bessel Functions
\[H_v(x) - Y_v(x) = \sum_{k=0}^{\infty} \frac{(-1)^k \Gamma(k+v+\frac{1}{2})}{\sqrt{\pi} \Gamma(\frac{1}{2}) k!} \left(\frac{x}{2}\right)^{2k-v}\]

Difference from Neumann function

Special Values

Important Values
H₀(0) = 0 H₁(0) = 0 H₀(π) ≈ 0.90
Behavior at x = 0
\[H_v(0) = 0\]

For all v > -1 (regular at origin)

Asymptotic Behavior
\[H_v(x) \sim Y_v(x) + \frac{2}{\pi x}\]

For large x (approaches Yv)

Symmetry Property
H-v-1(x) = (-1)^⌊v⌋ Hv(x)

For integer v

Application Areas

Forced oscillations, inhomogeneous wave equations, excitation problems in physics.

Struve Function Hv(x)

On this page, the Struve function Hv(x) is calculated. In mathematics, the Struve functions Hα(x) are solutions y(x) of the inhomogeneous Bessel differential equation.

Mathematical Definition

The Struve function is the particular solution of the inhomogeneous Bessel differential equation:

\[x^2y'' + xy' + (x^2 - v^2)y = \frac{4\left(\frac{x}{2}\right)^{v+1}}{\sqrt{\pi}\Gamma\left(v+\frac{1}{2}\right)}\]

Properties

  • Regular at origin: Hv(0) = 0 for v > -1
  • Asymptotic behavior: Hv(x) ~ Yv(x) for large x
  • Physical meaning: Describes forced oscillations
  • Series expansion: Converges for all finite x
  • Recurrence formulas: Modified Bessel recurrence
  • Integral representation: Available for v > -1/2

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