Calculate Modified Struve Function

Online calculator for computing the Modified Struve function Lv(x)

Modified Struve Function Calculator

Hyperbolic Modification

The Lv(x) or modified Struve function shows exponential growth and is analogous to the modified Bessel function.

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

Modified Struve Function Curve

Mouse pointer on the graph shows the values.
The modified Struve function shows exponential growth instead of oscillation.

What makes the modified Struve function special?

The modified Struve function arises through hyperbolic transformation of the classical Struve function:

  • Hyperbolic ODE: x²y'' + xy' - (x² + v²)y = f(x)
  • Exponential growth: Lv(x) ~ e^x/√(2πx) for large x
  • Relation to Iv: Analogous to modified Bessel functions
  • Physical meaning: Damped/amplified systems
  • Heat conduction: Inhomogeneous problems with sources
  • Application: Diffusion and transport processes

Hyperbolic vs. Oscillating Bessel Equations

The modified Struve function solves the hyperbolic version of the inhomogeneous Bessel equation:

Classical Struve Hv
\[x^2y'' + xy' + (x^2 - v^2)y = f(x)\]

Oscillating solutions

Modified Struve Lv
\[x^2y'' + xy' - (x^2 + v^2)y = f(x)\]

Exponentially growing solutions

Formulas for the Modified Struve Function

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

Power series expansion (positive terms!)

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

For large x (relation to modified Bessel function)

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

Modified recurrence with sign change

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

Integral form with hyperbolic sine

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

Difference from modified Bessel function

Special Properties

Important Values
L₀(0) = 0 L₁(0) = 0 L₀(1) ≈ 0.276
Behavior at x = 0
\[L_v(0) = 0\]

For all v > -1 (regular at origin)

Exponential Growth
\[L_v(x) \sim \frac{e^x}{\sqrt{2\pi x}}\]

For large x (dominant behavior)

Symmetry Property
L-v-1(x) = Lv(x)

For integer v

Application Areas

Diffusion processes, heat conduction with sources, amplified systems, hyperbolic PDEs.

Formula for the Modified Struve Function Lv(x)

On this page, the Modified Struve function Lv(x) is calculated. In mathematics, the Struve functions are solutions to the inhomogeneous Bessel differential equation.

Mathematical Definition

The modified Struve function is the particular solution of the modified (hyperbolic) 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: Lv(0) = 0 for v > -1
  • Exponential growth: Lv(x) ~ e^x/√(2πx) for large x
  • Positive series coefficients: Monotonically increasing function
  • Hyperbolic nature: Solution of the modified ODE
  • Relation to Iv: Asymptotically similar to modified Bessel functions
  • Physical meaning: Amplified/damped systems with excitation

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