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Values of the irregular Coulomb wave function $G_\ell(\eta,\rho)$
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Numbers
$\ell$
$\eta$
$\rho$ 
$G_\ell(\eta,\rho)$
0
2
4:
1.397483426793015884269983643065655653088975198920867234392818405587707866331838282163337559489721121
0
2
5:
0.7944454706418513509053749418046889800570774878195826388290490256019634138182218243121653849574847736
0
2
1:
9.800335768445246642042903222976723495201498284285333654700118891807324291589061392330813405389781683
0
0
1:
0.5403023058681397174009366074429766037323104206179222276700972553811003947744717645179518560871830893
0
0
1/2:
0.8775825618903727161162815826038296519916451971097440529976108683159507632742139474057941840846822584
0
0
2:
-0.4161468365471423869975682295007621897660007710755448907551499737819649361240791690745317778601691404
0
0
3:
-0.9899924966004454572715727947312613023936790966155883288140859329283291975131332204282944793556926022
0
0
5:
0.2836621854632262644666391715135573083344225922522159449303590665861514567673827022861769816683445732
0
0
8:
-0.1455000338086135258688413818311946826093196855455212253509644786956803082878734006162861713726358184
0
1
1:
2.043097162103538072356309506931951984052244871976324140280237166060199722837999021735667987301435239
0
1
1/2:
3.106006927954887514047275971035543079826411411706030804345912460260699931339096642390538716814869952
0
1
2:
1.275778784768276589181585501253655475557796992466478490564193262125858923266129513087291450564855083
0
1
3:
0.6270395148892855262169606659419037770708691306346391045982978543792658745665115599586227869459462027
0
1
5:
-0.8984143590920205486928740511191748327465792731706679275085577589873572672272204188230981117116857888
0
1
8:
0.2911426012378539717511012000279962959141558958761296394108605194381403416537682008297565951739472448
0
-1
1:
-0.5673621513069320177954406861439467585606372967538745224307564951867086753318296064621899277070065607
0
-1
1/2:
0.06911237251094593239316518174442452582698374295499063595274958658497580293790565612148995370285345904
0
-1
2:
-0.6047081229639326193394252295903906301905486613374232512187946352350525519964429224695527984206774499
0
-1
3:
0.4524863921628617039287024553150667386839865497978542051039829234657501272981810935580376441311798876
0
-1
5:
0.1395925277618273183713507245866557674303999732732088439259138835955884980263341809635744703127805290
0
-1
8:
0.1324013567958338579507617426962852660959119845044717567743277730183385708777972438804054760157580657
0
1/2
1:
1.197486970798485546289462250046922453782204014437122729619187315396780683744638564371938896879380368
0
-1/2
1:
-0.1366043683725464235676693694411497622520912515204662622471856340773930280113869500496883790142717180
0
-2
1:
-0.6396339014848806951521732720089787249293031180206184538980282589058430042367856005309164920956434556
1
0
$\sqrt{2}$:
1.098034790044616931457621673741419494484492773203776629973275613421658873473743404314090023373031836
1
0
1:
1.381773290676036224053438929073275603354873481416293293342848965373010799165711433466591599630235785
1
2
$2+\sqrt{6}$:
1.398790881777736054889244402577010378467216985436026428964147689724114748596160712753406900653252223
1
2
5:
1.081482760400010974966334352132403848548443519855765966810827627305751464868057196969288017852090439
1
1
1:
3.782874510510568188143165093566824567332144373021811405553217404863469927158962693544225143922714873
1
1
$1+\sqrt{3}$:
1.279319915700653654185639509421813804633010784272088335844673509653240831972936444784441950687640580
1
1
2:
1.765219240339179457333742228503363572296828118024706968650654578188433891836090186273515259272376549
1
-1
1:
0.6802666341833721116102138484334407518798526209263456755727671059924997511435136499540806108503141323
1
-1
$-1+\sqrt{3}$:
0.8752308885849067029062348866957642791151623653666524277746510277003220720101896833729686017684375670
1
-1
2:
-0.3457464103835453997219160887246772901896285122854106707592888980229928860760243430820832418548315472
2
0
$\sqrt{6}$:
1.166533155662417060952449180289157614670101598951124307782002774934812859565311284109866619812149653
2
0
1:
3.605017566159968954759380179776850206332310023630957652358449640737932002722662535881822942803524266
2
2
$2+\sqrt{10}$:
1.405511327439484975128328228670995621841723806383259859020606197243006189594627406443013639487383129
2
2
5:
1.496877074998357979395992654859848387486288040027978361923528005603136699119914824317969327980856709
2
1
1:
12.64143999522579871232479759208836567080484864417230303123038547612827769178465435019034486198960706
2
1
$1+\sqrt{7}$:
1.300049761730886974371592628030555193072470248695903253769654141688796934643788080565096232232452431
2
-1
1:
1.630336124636594500754064590072031403373321233575084887052011483248774056054015004167334065491362195
2
-1
$-1+\sqrt{7}$:
1.018467294445741818665685566247688712325769669133781781669463823373979559802722994094587687941382354
3
0
$\sqrt{12}$:
1.221275053471756383494608391687610831784907227076868443841640678408373710836042414182184403847530565
3
0
1:
16.64331454012380854974346196981097542830667663673849496844939923831664921444760124594252311438738555
3
2
6:
1.417653661460542689974739482834097239001920368536942446875981595157152180754264179695376439279642713
3
2
5:
2.048694713777195592942384636612149848890630178081008703408405290702015596033207036444701753170042215
3
1
1:
65.94520642806816588511738028797163086329248217119323699209075199252583891069103151845641157746220003
3
1
$1+\sqrt{13}$:
1.326787668183029361085796346548739441453247131461219429066352138628924313777807854459069132127119880
3
-1
1:
5.722937656810725333608846923835988097745564881776942074556710749999204435092973005347926867733624985
3
-1
$-1+\sqrt{13}$:
1.108550991499070229120380513296423086645877308831225421212061707936494329980540040074335806089762363
Definition
The irregular Coulomb wave function $G_\ell(\eta,\rho)$ is the real solution of Coulomb's equation (1) with $G_\ell(\eta,\rho)=\cos\theta_\ell(\eta,\rho)+o(1)$ as $\rho\to+\infty$, where (2) fixes the phase [1] [2].
Parameters
$\ell$
—   angular momentum ($\ell\in\mathbb{Z}_{\geq0}$)
$\eta$
—   Coulomb parameter ($\eta\in\mathbb{Q}$)
$\rho$
—   dimensionless radius ($\rho>0$)
Formulas
(1)
$\frac{d^2w}{d\rho^2}+\left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}{\rho^2}\right)w=0$ [1].
(2)
$\theta_\ell(\eta,\rho)=\rho-\eta\log(2\rho)-\ell\pi/2+\sigma_\ell(\eta)$, where $\sigma_\ell(\eta)=\operatorname{Im}\log\Gamma(\ell+1+i\eta)$ is continuous in real $\eta$ and is zero at $\eta=0$ [1].
(3)
$G_\ell(\eta,\rho)=\operatorname{Re}\!\left[e^{i\theta_\ell(\eta,\rho)}(-2i\rho)^{\ell+1+i\eta}U(\ell+1+i\eta,2\ell+2,-2i\rho)\right]$, where $U$ is Tricomi's confluent hypergeometric function. The power and $U$ use their principal branches; $\theta_\ell$ is defined in (2) [1].
(4)
$G_\ell(0,\rho)=-\sqrt{\pi\rho/2}\,Y_{\ell+1/2}(\rho)$, where $Y_\nu$ is the Bessel function of the second kind. In particular, $G_0(0,\rho)=\cos\rho$ [3].
(5)
For $j\geq1$, let $R_j=\sqrt{1+\eta^2/j^2}$ and $S_j=j/\rho+\eta/j$. Then $R_\ell G_{\ell-1}-(S_\ell+S_{\ell+1})G_\ell+R_{\ell+1}G_{\ell+1}=0$ for $\ell\geq1$, with common arguments $(\eta,\rho)$ suppressed [4].
(6)
With $R_j,S_j$ from (5), $\partial_\rho G_\ell=S_{\ell+1}G_\ell-R_{\ell+1}G_{\ell+1}$ for $\ell\geq0$, and $\partial_\rho G_\ell=R_\ell G_{\ell-1}-S_\ell G_\ell$ for $\ell\geq1$ [4].
(7)
$G_\ell\,\partial_\rho F_\ell-F_\ell\,\partial_\rho G_\ell=1$, where $F_\ell$ is the regular Coulomb wave function [1].
Comments
(8)
This is the DLMF normalization with $H_\ell^\pm=G_\ell\pm iF_\ell$, where $F_\ell$ is the regular Coulomb wave function [1]. The asymptotic condition fixes both the sign of $G_\ell$ and the possible addition of a multiple of $F_\ell$. Values retain their signs.
(9)
Positive $\eta$ describes repulsion and negative $\eta$ attraction. In scattering, $\rho=kr$, where $r$ is the separation and $k$ is the wave number [5]. The orders $\ell=0,1,2,3$ are the s, p, d and f partial waves.
(10)
It preserves the existing free-wave turning points, handbook benchmarks and their turning-point companions. The s, p, d and f waves at $\rho=1$ add $\eta=0,\pm1$, with the positive outer turning points also included for $\ell=1,2,3$ at $\eta=\pm1$. The s-wave references add $\rho=1/2,2,3,5,8$ at $\eta=0,\pm1$ and $\rho=1$ at $\eta=\pm1/2,\pm2$; the p wave also uses $\rho=2$ at $\eta=\pm1$. Zero coupling is the free-wave degeneration. The signed half, unit and double couplings are bounded exploratory references in the DLMF normalization, not fixed physical constants. The sparse radii and low partial-wave cutoff are practical choices, and a turning point is intrinsic only conditional on its chosen coupling. [12] [13] [14]
(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: The free case eta=0 is an intrinsic degeneration. For the p, d and f partial waves ell=1,2,3, the centrifugal turning radius is sqrt(ell(ell+1)). The s-wave free case has no positive turning radius and is not included in that group. The handbook worked examples include ell=0,1,2,3 at eta=2, rho=5 as reproducibility benchmarks, not a measured or mathematically distinguished coupling. At that same reference coupling the outer turning radius is 2+sqrt(4+ell(ell+1)), intrinsic conditional on eta=2. A turning point means a zero of the coefficient in the reduced wave equation, not a stationary point of F or G. No nonzero eta is claimed physically fixed. [8]. [9] [10] [11]
(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
ell, eta = 1, 0
rho = mp.sqrt(ell * (ell + 1))
print(mp.nstr(mp.coulombg(ell, eta, rho), 100))
Links
[12]
DLMF 33.2
[13]
DLMF 33.5
[14]
DLMF 33.22
Similar tables
Values of the regular Coulomb wave function —   the companion solution $F_\ell$ satisfies (7) with $G_\ell$
Values of Bessel functions of the second kind —   half-integer orders give the free case $\eta=0$ through (4)
Values of Tricomi's confluent hypergeometric function —   $U$ gives $G_\ell$ through (3); the linked table holds real parameters and positive real arguments
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it preserves the existing free-wave turning points, handbook benchmarks and their turning-point companions. The s, p, d and f waves at $\rho=1$ add $\eta=0,\pm1$, with the positive outer turning points also included for $\ell=1,2,3$ at $\eta=\pm1$. The s-wave references add $\rho=1/2,2,3,5,8$ at $\eta=0,\pm1$ and $\rho=1$ at $\eta=\pm1/2,\pm2$; the p wave also uses $\rho=2$ at $\eta=\pm1$. Zero coupling is the free-wave degeneration. The signed half, unit and double couplings are bounded exploratory references in the DLMF normalization, not fixed physical constants. The sparse radii and low partial-wave cutoff are practical choices, and a turning point is intrinsic only conditional on its chosen coupling)
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.

more

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.

digits: 100 (Unknown key)


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