Values of the modified Struve function $\mathbf{L}_\nu(x)$

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Numbers

$\nu$

$x$

$\mathbf{L}_\nu(x)$

0.7102431859378908887385266778116507449444099583297031981230151655739691156882381809773540391579117258

1/2:

0.3272406993941807802475218143697526013249599983276467039800203277942434941612902357400114406001067032

1.937433757991445661161521326319693095686482529772495335358244701075522654352853696364746933831895688

4.648685731753710282598935769061639910923351238852300918232542604589111528137790184294552446159375729

27.10591712655814655322253488178511411832852843862841621443402639646847522074929510289108766028968053

427.4830347130955659684785419034990276600709411724531929317205241146519112272799317812809950566015203

0.2267643810558086368261753538677918847361832955676687095737719815374595587599634353384819291648013737

1/2:

0.05394218262352266329233324711166140172769854724645848875528433177768845699476837411978483823558464379

1.102759787367715817589721650181908914811889696943731540829103646297867070648572235328031286632879281

3.396612947474775997751454161593429909992653904405835989738612020939423300702031647964535431124464143

23.72821578040828244582709544174408094553749528639606432144708483918569428100611959050728406077193052

399.2470913872569767243187488725777172921478814420467418663025269390845424503485882574378444355723965

0.04450783303707983406099761891271449275943050620709051397858141042052093465600926351357177713997153058

0.006291730749650544377149207984263417155938698541851600660068381439670007433465971926831139867247761756

0.0006944030880567454830972752596096331748665189277730453039742705802572025262905163301044121168181758143

1/2

0.4333156537901020906259996225857388668956874487678961059711137653576900787515395272455748988118907898

1/2

1.558402036629880906868040844170805355275075030324819532014648653133900936400331887783237575453884640

1/2

4.177098891899722153709735552969182767831135537748051558296065688484478298806581825694150578488146055

1/2

26.12312694107539649757170348530765125212182101705424329562852806331331278691578930115127063832361290

3/2

0.1054169540539528781513172018712726324602430728988563515102636745183963607243954799857875372312671174

5/2

0.01732922152788528668706150198832550239599357228009338702384128438483651510352762766289920364126270352

Definition

The table gives the modified Struve function $\mathbf{L}_\nu(x)$ [1], with the normalization in (1), for rational order $\nu\geq0$ and real argument $x\geq0$.

Parameters

$\nu$

— order ($\nu\geq0$)

$x$

— argument ($x\geq0$)

Formulas

(1)

$\displaystyle\mathbf{L}_\nu(x)=\left(\frac{x}{2}\right)^{\nu+1}\sum_{k=0}^{\infty}\frac{(x/2)^{2k}}{\Gamma(k+3/2)\Gamma(k+\nu+3/2)}$, where $\Gamma$ is the gamma function. The positive real power $(x/2)^{\nu+1}$ at $x>0$ specifies the principal value [1].

(2)

For $x>0$ and $\nu\geq0$, $\mathbf{L}_{\nu-1}(x)-\mathbf{L}_{\nu+1}(x)=\dfrac{2\nu}{x}\mathbf{L}_\nu(x)+\dfrac{(x/2)^\nu}{\sqrt{\pi}\Gamma(\nu+3/2)}$ [2]. In particular, $\nu=0$ gives $\mathbf{L}_{-1}(x)=2/\pi+\mathbf{L}_1(x)$.

(3)

For $x>0$, $\mathbf{L}_{1/2}(x)=\sqrt{\dfrac{2}{\pi x}}(\cosh x-1)$ and $\mathbf{L}_{3/2}(x)=\sqrt{\dfrac{2}{\pi x}}\left(\sinh x-\dfrac{\cosh x-1}{x}-\dfrac{x}{2}\right)$. These are elementary half-integer cases of the modified Struve function [2].

(4)

For $x>0$ and real $\nu>-1/2$, $\displaystyle\mathbf{L}_\nu(x)=\dfrac{2(x/2)^\nu}{\sqrt{\pi}\Gamma(\nu+1/2)}\int_0^{\pi/2}\sinh(x\cos\theta)(\sin\theta)^{2\nu}\,d\theta$ [3].

(5)

For $x>0$, $\displaystyle\int_0^x I_0(t)\,dt=\dfrac{\pi x}{2}\left[I_0(x)\left(\dfrac{2}{\pi}+\mathbf{L}_1(x)\right)-I_1(x)\mathbf{L}_0(x)\right]$, where $I_\nu$ is the modified Bessel function of the first kind [4].

(6)

For $x>0$ and real $\nu\geq0$, $\mathbf{L}_\nu(x)=-i e^{-\nu\pi i/2}\mathbf{H}_\nu(ix)$, where $i^2=-1$ and $\mathbf{H}_\nu$ is the Struve function on its principal branch [1].

Comments

(7)

The continuous value at the origin is $\mathbf{L}_\nu(0)=0$ for $\nu\geq0$, as follows from the defining series.

(8)

The function $\mathbf{L}_{-1}$ that occurs in the Bessel-integral relation is expressed using $\mathbf{L}_1$ by (2), giving (5).

(9)

The bold symbol $\mathbf{L}_\nu$ denotes the modified Struve function. Its ordinary counterpart is the Struve function $\mathbf{H}_\nu$, related by rotating the argument as in (6).

(10)

It preserves all existing values and uses the integral-representation pair of orders $0,1$ at $x=0,1/2,1,2,3,5,8$, with orders $2,3,4,1/2,3/2,5/2$ at $x=1$ and order $1/2$ also at $x=2,3,5$. The integer-order recurrences and elementary half-order formulas give structural breadth. The positive radii are sparse, bounded exploratory references for the sine or hyperbolic-sine integrals, not a mesh or documented demand for every argument. Exact origin values remain exact. [7] [8] [9]

(11)

The reference inputs and finite cutoffs are explicit choices rather than a completeness or application-demand claim. The attached balanced-selection.json specifies every argument exactly. Existing entries retain their identifiers, decimal strings and annotations.

(12)

The retained pre-extension subfamily has the following argument definitions: Every coefficient of the defining modified-Struve series is positive for nonnegative order and positive argument, and differentiation leaves strictly positive terms. There are no positive zeros or stationary points to use. The origin and just one explicit reference argument x=1 are retained for orders 0 and 1, the pair occurring in the integral of I0. The unit argument is a modest reference sample, not a mathematically distinguished argument or a physically fixed parameter. [5]. [6]

(13)

The attached generate.py forwards to balanced_generate.py, the standalone generator for the entire current selection. Its default offline check recomputes both new and retained values. The previous entry point is preserved byte-for-byte as pre-balanced-generator.py and in revision history. Earlier supporting sources and checks are preserved as provenance for their original subfamilies; balanced-provenance.json identifies their source hashes and the retained website revision.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 150
print(mp.nstr(mp.struvel(0, 1), 100))

Links

[1]

DLMF 11.2.2: Modified Struve function

[2]

DLMF 11.4: Basic properties of Struve and modified Struve functions

[3]

DLMF 11.5.6: Integral representation of the modified Struve function

[4]

DLMF 10.43.2: Modified Bessel integrals in terms of modified Struve functions

[5]

Argument selection: 11.2.E2

[6]

Argument selection: 10.43.E2

[7]

DLMF 11.2

[8]

DLMF 11.4

[9]

DLMF 11.5

Similar tables

Values of the Struve function $\mathbf{H}_\nu(x)$ — $\mathbf{H}_\nu$ is the ordinary Struve function; argument rotation gives $\mathbf{L}_\nu$ by (6)

Values of modified Bessel functions of the first kind $I_\nu(x)$ — $I_\nu$ solves the homogeneous modified Bessel equation associated with the equation for $\mathbf{L}_\nu$ and occurs in (5)

Values of modified Bessel functions of the second kind $K_\nu(x)$ — $K_\nu$ is the other standard solution of that homogeneous modified Bessel equation

Values of Bessel functions of the first kind $J_\nu(x)$ — $J_\nu$ solves the homogeneous Bessel equation associated with the ordinary Struve function $\mathbf{H}_\nu$

Values of Bessel functions of the second kind $Y_\nu(x)$ — $Y_\nu$ is the second standard solution of the homogeneous Bessel equation associated with $\mathbf{H}_\nu$

Values of the Kelvin function $\operatorname{ber}_\nu(x)$ — $\operatorname{ber}_\nu$ is the real part of a Bessel function at a rotated argument, another family of cylinder-function values

Values of the Kelvin function $\operatorname{bei}_\nu(x)$ — $\operatorname{bei}_\nu$ is the imaginary part of the rotated Bessel value whose real part defines $\operatorname{ber}_\nu$

Values of the Kelvin function $\operatorname{ker}_\nu(x)$ — $\operatorname{ker}_\nu$ is the real part of a phased modified Bessel function at a rotated argument

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it preserves all existing values and uses the integral-representation pair of orders $0,1$ at $x=0,1/2,1,2,3,5,8$, with orders $2,3,4,1/2,3/2,5/2$ at $x=1$ and order $1/2$ also at $x=2,3,5$. The integer-order recurrences and elementary half-order formulas give structural breadth. The positive radii are sparse, bounded exploratory references for the sine or hyperbolic-sine integrals, not a mesh or documented demand for every argument. Exact origin values remain exact)

How they were obtained:

Retained-entry provenance (unchanged numerical data): Arb ball arithmetic encloses the function and its argument uncertainty at 773 bits for 100 requested significant digits. Critical arguments are recomputed from their defining rational intervals, with opposite endpoint signs, a Gronwall bound proving the second derivative has a fixed sign throughout the interval, and a separately certified narrow bracket. Turning points use exact integers and enclosed square roots. Exact origin values remain exact integers. Every stored interval contains an independently recomputed 900-bit enclosure and agrees with mpmath at 160 and 210 decimal digits. Kelvin and Struve values additionally agree with numerical quadrature of elementary integral representations; these agreement checks do not establish the proven bounds. Coulomb checks include the unit Wronskian, the vanishing turning-point coefficient, and the free-case Bessel identities. Nonzero controls and a deliberately displaced value test the comparisons. See checks-evidence.json for per-entry measurements. Earlier grid-specific checks and sources remain historical evidence and do not certify this selection.

Balanced extension: balanced_generate.py recomputes the entire attached selection, including retained values. New values use Arb ball arithmetic with exact rational inputs or enclosed algebraic and hyperbolic inputs at 773 working bits for 100 significant digits. Exact origin anchors are stored as integers. The attached critical-argument source is preserved and hash-identified; its certified argument enclosures are propagated through the evaluation. All old numerical strings, higher precision and manual entry annotations are preserved. balanced-checks.json records full independent higher-precision Arb recomputation and independent mpmath evaluations at 160 and 210 digits for every new value, with finite-width controls and family-specific identities or integral checks. The generator does not claim authorship of retained data; their prior supporting attachments remain unchanged, and the previous entry point is preserved as pre-balanced-generator.py.