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

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Numbers

$\nu$

$x$

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

$c_{1}$:

0.7909687719198929225138069805538998077694051064800173282438079358644946229614482078928190764412733063

$c_{2}$:

-0.2268761587374463267889574552506960608050668645285051414166588755501960481825629264113734184337629129

$c_{3}$:

0.3447409745070508619983657063601800460052249270305289938971778388295397689030089868822152782041583495

$c_{4}$:

-0.1786917666761634960366846841666848876594674561524079967209337778067421683121396549380169737171101999

$c_{5}$:

0.2491145753273138953561737302041158520798811355701787761652515838425709132388011463498560719865089876

$c_{6}$:

-0.1525619773841306370877494109598955192175744209539345853467558929600080885368186950088054846618822633

1/2:

0.3095559145837547181640731567486780257723048810891822447101994837939990326768417617820039224303374548

0.5686566270482879509864228863223532743026778272555587381111952989827924343658033118878157159186508343

0.7908588495080958925516676824833390825127057075765193232166240178714817966724700633983008468994556730

0.5743061488143983979756175433066279369885366610067159425630180900885167336784432276852621954975802937

-0.1852168157766848901057186921439789100378565572275162151406693225220514033573363973771441454875085838

0.3019881117101377668915517163203120783455477726750176555664902628556142042595817253091472559708342781

$c_{1}$:

1.094735275025846542473444436836938775717366353215959125090076073625455931395478232268847874885789703

$c_{2}$:

0.3459791306973964822106811924522236099023229710431721367936678050563785543733401937030965154437260278

$c_{3}$:

0.8935857641423305384001203068273204882656225181351675818893335026392184808473779810952044736579690161

$c_{4}$:

0.4211941513488293225894222592668124725748557389763436940170773442501046283049613423493573237203991540

$c_{5}$:

0.8357781775068627420453999853511403360206525533559774987247269423902682302073622862118314699858793421

$c_{6}$:

0.4579686068429420508062404906046690888537048326897746431807366120183128345273134831638535083765412840

1/2:

0.05217374424234107037559789471257600041045499143139962824212401661462117263931690624083194397303856788

0.1984573362019443989353381326480237720460574512008611700027443444545899117388418510243473814725107408

0.6467637282835621171228332858530973836621359647507871419997160136486246856334351083412552529172976619

1.020109569186450360812823159076296095691138265063217900005751115798995352584333756301650648720362482

0.8078119457940644415036446898345495769055795689964175101063283476363927604280116938414924410446071380

0.4881160461214211402672246421251974779519725752519392378973258185417650202750671676558181744484351357

0.04046463614479462790943173013704675250204993613343886689118318200491645262418243694769745071470809533

0.005842526535072868907424458132833734504664865530349350561366341980781711460348306123806102123989130054

0.0006535685167624078501201144074800111748710530768651011912117210820175036668303154176196878488934692245

1/2

0.3667856927844892763593711525702303368636612516412524106794800609884366562532374559515787489453240531

1/2

0.7989752939540047561221141353541837957980771385359496808215658612760263126623797470627506952274063638

1/2

0.9167076867564138178671594549208015901037289345869471980845846661478014838390940474153858356109092581

1/2

0.2556071140454458233427330975215717863690821652304881890596812597450560659842010849885831693627043430

3/2

0.09433126604411886388295320046414512479387906603684115508602707316664734478735143091867458211563434287

5/2

0.01594367544822548477447496380580050463694060426050471899508261592916985958364229642975808087840570964

Definition

The table gives the Struve function $\mathbf{H}_\nu(x)$ [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)

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

(2)

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

(3)

For $x>0$, $\mathbf{H}_{1/2}(x)=\sqrt{\dfrac{2}{\pi x}}(1-\cos x)$ and $\mathbf{H}_{3/2}(x)=\sqrt{\dfrac{2}{\pi x}}\left(\dfrac{x}{2}+\dfrac{1-\cos x}{x}-\sin x\right)$. The second identity follows from (2) and $\mathbf{H}_{-1/2}(x)=\sqrt{2/(\pi x)}\sin x$ [2].

(4)

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

(5)

For $x>0$, $\displaystyle\int_0^x J_0(t)\,dt=\dfrac{\pi x}{2}\left[J_0(x)\left(\dfrac{2}{\pi}-\mathbf{H}_1(x)\right)+J_1(x)\mathbf{H}_0(x)\right]$, where $J_\nu$ is the Bessel function of the first kind [4].

Comments

(6)

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

(7)

The order $-1$, which is eliminated in deriving (5), is determined by $\mathbf{H}_{-1}(x)=2/\pi-\mathbf{H}_1(x)$, the case $\nu=0$ of (2).

(8)

The bold symbol $\mathbf{H}_\nu$ denotes the Struve function; the Hankel functions are written $H_\nu^{(1)}$ and $H_\nu^{(2)}$ [1].

(9)

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. [8] [9] [10]

(10)

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.

(11)

For $\nu=0$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(2025/1024,1013/512)$, $(5619/1024,1405/256)$, $(4385/512,8771/1024)$, $(12051/1024,3013/256)$, $(1905/128,15241/1024)$, $(18487/1024,2311/128)$, respectively. For $\nu=1$, $c_1,\ldots,c_6$ are the unique zeros of the derivative in $(929/256,3717/1024)$, $(7119/1024,445/64)$, $(10361/1024,5181/512)$, $(13607/1024,1701/128)$, $(16833/1024,8417/512)$, $(20061/1024,10031/512)$, respectively. These labels identify arguments, not the values at them.

(12)

The retained pre-extension subfamily has the following argument definitions: The order-zero and order-one Struve functions occurring in the integral of J0 are evaluated at six isolated critical arguments each and at their continuous origin values. Each critical argument is the unique derivative zero in its stated rational interval. The labels enumerate this finite selection, not a claimed global zero count. No stationary points are assumed to exist for other orders. [5]. [6] [7]

(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
left = mp.mpf('2025') / 1024
right = mp.mpf('1013') / 512
argument = mp.findroot(lambda point: mp.diff(
    lambda value: mp.struveh(0, value), point), (left, right))
print(mp.nstr(mp.struveh(0, argument), 100))

Links

[1]

DLMF 11.2: Definitions

[2]

DLMF 11.4: Basic Properties

[3]

DLMF 11.5.1: Integral representation of the Struve function

[4]

DLMF 10.22.2: Bessel integrals in terms of Struve functions

[5]

Argument selection: 11.2

[6]

Argument selection: 11.4

[7]

Argument selection: 10.22.E2

[8]

DLMF 11.2

[9]

DLMF 11.4

[10]

DLMF 11.5

Similar tables

Values of Bessel functions of the first kind $J_\nu(x)$ — $J_\nu$ solves the homogeneous Bessel equation associated with Struve's equation and occurs in (5)

Values of Bessel functions of the second kind $Y_\nu(x)$ — $Y_\nu$ is the other standard solution of the homogeneous Bessel equation associated with Struve's equation

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

Values of modified Bessel functions of the second kind $K_\nu(x)$ — $K_\nu$ is the second standard solution of the homogeneous equation associated with the modified Struve function $\mathbf{L}_\nu$

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 same rotated Bessel value that defines $\operatorname{ber}_\nu$

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.