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{"Title": "Values of the irregular Coulomb wave function $G_\\ell(\\eta,\\rho)$", "Definition": "The irregular Coulomb wave function $G_\\ell(\\eta,\\rho)$ is the real solution of Coulomb's equation CITE{formula-equation} with $G_\\ell(\\eta,\\rho)=\\cos\\theta_\\ell(\\eta,\\rho)+o(1)$ as $\\rho\\to+\\infty$, where CITE{formula-phase} fixes the phase CITE{DLMFDefinition} CITE{DLMFAsymptotic}.", "Parameters": {"ell": {"type": "Z", "title": "angular momentum", "display": "$\\ell$", "constraints": "$\\ell\\in\\mathbb{Z}_{\\geq0}$"}, "eta": {"type": "Q", "title": "Coulomb parameter", "display": "$\\eta$", "constraints": "$\\eta\\in\\mathbb{Q}$"}, "rho": {"type": "Q", "title": "dimensionless radius", "display": "$\\rho$", "constraints": "$\\rho\\in\\mathbb{Q}_{>0}$"}}, "Links": {"DLMFDefinition": {"title": "DLMF 33.2: Definitions and Basic Properties", "url": "https://dlmf.nist.gov/33.2"}, "DLMFAsymptotic": {"title": "DLMF 33.10: Limiting Forms for Large Radius", "url": "https://dlmf.nist.gov/33.10"}, "DLMFLimits": {"title": "DLMF 33.5: Limiting Forms and the Free Case", "url": "https://dlmf.nist.gov/33.5"}, "DLMFRecurrence": {"title": "DLMF 33.4: Recurrence Relations and Derivatives", "url": "https://dlmf.nist.gov/33.4"}, "DLMFPhysical": {"title": "DLMF 33.22: Particle Scattering and Atomic and Molecular Spectra", "url": "https://dlmf.nist.gov/33.22"}, "Mpmath": {"title": "mpmath: Coulomb wave functions and numerical examples", "url": "https://mpmath.readthedocs.io/en/stable/functions/bessel.html#coulomb-wave-functions"}, "ASExample": {"title": "Abramowitz and Stegun, Handbook of Mathematical Functions, page 543, Example 2", "url": "https://personal.math.ubc.ca/~cbm/aands/page_543.htm"}}, "Tags": ["special functions", "special values", "physics"], "Data properties": {"type": "R", "rigour": "proven", "complete": "no", "complete-note": "it holds the selected reference couplings $\\eta=0,\\pm1/2,\\pm1,\\pm2$ for the s, p, d and f partial waves $\\ell=0,1,2,3$, at $\\rho=1/2,1,2$ around the reduced wavelength, with $\\rho=5$ additionally included at $\\eta=2$ for comparison with published handbook values", "rigour details": "The attached generator evaluates the DLMF representation using HREF{Values_of_Tricomi's_confluent_hypergeometric_function}[Tricomi's confluent hypergeometric function $U$] in Sage/Arb complex-ball arithmetic. It computes $G_\\ell=(H_\\ell^++H_\\ell^-)/2$ with exact rational arguments and accepts only finite enclosures whose imaginary parts contain zero. At 515 working bits, the minimum measured relative accuracy was 496 bits and every real-ball radius was less than $2.711\\times10^{-149}$. Each written 100-significant-digit decimal denotes an interval extending one unit in its last place in either direction; this interval contains the entire computed ball.\n\nAll 88 values agree with an independent mpmath implementation at both 150 and 200 decimal digits CITE{Mpmath}. The attached checks test the differential equation, the Wronskian with HREF{T510}[the regular Coulomb wave function], both derivative recurrences where applicable, and the order recurrence. Derivatives are evaluated analytically using shifted parameters of $U$, and the regular solution is evaluated using a hypergeometric series with a separate product normalization. The direct DLMF phase representation is compared with the generator's equivalent gamma-ratio representation on every entry. Enclosures must be finite and narrow before an identity is accepted.\n\nThe 12 free-case entries agree with HREF{Values_of_Bessel_functions_of_the_second_kind}[the Bessel function of the second kind $Y$], including three order-zero comparisons with cosine. At $(\\eta,\\rho)=(2,5)$, four values and their derivatives agree with Abramowitz and Stegun's worked example within $10^{-4}$ in absolute value CITE{ASExample}. Its rounded input and intermediate values do not support every printed final digit. The three higher-order values included here and their derivatives agree with the higher-precision mpmath documentation examples within one unit in the last printed place. Nonzero trigonometric and Wronskian controls, a deliberately reversed sign, and an exact decimal-interval control test the checking methods."}, "Display properties": {"number-header": "$G_\\ell(\\eta,\\rho)$"}, "Numbers": {"0": {"0": {"1/2": "0.8775825618903727161162815826038296519916451971097440529976108683159507632742139474057941840846822584", "1": "0.5403023058681397174009366074429766037323104206179222276700972553811003947744717645179518560871830893", "2": "-0.4161468365471423869975682295007621897660007710755448907551499737819649361240791690745317778601691404"}, "1/2": {"1/2": "1.516673949995264952866499014842669389991727335466316544813342411169801853620028288364836345038306616", "1": "1.197486970798485546289462250046922453782204014437122729619187315396780683744638564371938896879380368", "2": "0.5322112059077912322052031455506962799855619407755536896919776667100625556679452818138627431966265089"}, "-1/2": {"1/2": "0.4283099439614368633563452358943067631975566635099955398906380795185852016108625535586134133791995887", "1": "-0.1366043683725464235676693694411497622520912515204662622471856340773930280113869500496883790142717180", "2": "-0.8997769591031676373408824503694541145239517346838277929420864830215791018968105353970857974115164019"}, "1": {"1/2": "3.106006927954887514047275971035543079826411411706030804345912460260699931339096642390538716814869952", "1": "2.043097162103538072356309506931951984052244871976324140280237166060199722837999021735667987301435239", "2": "1.275778784768276589181585501253655475557796992466478490564193262125858923266129513087291450564855083"}, "-1": {"1/2": "0.06911237251094593239316518174442452582698374295499063595274958658497580293790565612148995370285345904", "1": "-0.5673621513069320177954406861439467585606372967538745224307564951867086753318296064621899277070065607", "2": "-0.6047081229639326193394252295903906301905486613374232512187946352350525519964429224695527984206774499"}, "2": {"1/2": "23.30264828037038918993596144263868006226600905558802816804070339496289284317073990254227668799281778", "1": "9.800335768445246642042903222976723495201498284285333654700118891807324291589061392330813405389781683", "2": "3.512383091309328645479476975768605274040175302786865529956646157741404392492984926033005054800451274", "5": "0.7944454706418513509053749418046889800570774878195826388290490256019634138182218243121653849574847736"}, "-2": {"1/2": "-0.3590146510969463504713138160642745323060239105980387268825510681956600211450055227220870744358297405", "1": "-0.6396339014848806951521732720089787249293031180206184538980282589058430042367856005309164920956434556", "2": "0.4725003083182785196118284397153128243460285148352173350850649225373922057102128224673962050894761413"}}, "1": {"0": {"1/2": 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"0.1601619109506731634055961726921740523094740020237065812074442786023568908765532115844597370815402936"}, "1": {"1/2": "9.098467985975935239176517897297798629593556883733931338591113502165945305750070466450477710867610549", "1": "3.782874510510568188143165093566824567332144373021811405553217404863469927158962693544225143922714873", "2": "1.765219240339179457333742228503363572296828118024706968650654578188433891836090186273515259272376549"}, "-1": {"1/2": "1.051512221354637385703240156390450378676741001044583978430606306705302705147884972689240777808229121", "1": "0.6802666341833721116102138484334407518798526209263456755727671059924997511435136499540806108503141323", "2": "-0.3457464103835453997219160887246772901896285122854106707592888980229928860760243430820832418548315472"}, "2": {"1/2": "64.18385277066010654459570939763077318099232256913048277779900041496697984647344575418525507560909594", "1": "19.32572348470850240649169801187841479928662279401459383908270785068990564987730709712357627478668676", "2": "5.069786005174830501530748783487625804595282102996973888614637450917405598121352000354478933992415033", "5": "1.081482760400010974966334352132403848548443519855765966810827627305751464868057196969288017852090439"}, "-2": {"1/2": "0.6670573971753702464226359487309885927849652512103318642123980175657287962440882065461941803316190761", "1": "0.1284759554300819837887014654957519687905202690893562105351640237757187286902134949923473482002143821", "2": "-0.7942591281405753062121095804772697022116607869176406976776825030603091364289034642896639974244170862"}}, "2": {"0": {"1/2": "12.52996141241931787891882501993555450039891737585078863410541923022866839774324241472152367303759446", "1": "3.605017566159968954759380179776850206332310023630957652358449640737932002722662535881822942803524266", "2": "1.467982849375308139843421856242807811494882649940721625257247010741942186342195109198822977852481356"}, "1/2": {"1/2": "25.44798528474060029321663479236851994930392179536207839135097172614711514836804079069060387643878185", "1": "6.372521952279470838783228169573723208300582736613797375528606279068094943694510836324815253645851798", "2": "2.038405608443596529585103593136651172054900445625476161665184588573988909520145579151186631387237825"}, "-1/2": {"1/2": "6.848245908940055607022928923042866774914502450999540259889168742908955937088238122891485623366345712", "1": "2.297587841625951233126449607711404951626242564980616431281654675442841830697067970908557056269831377", "2": "1.092480045902947188525914053224932354261087779013765471309399173053379500142522703515953173942214062"}, "1": {"1/2": "57.10555619500340024226479138236223259027294898271021346905495847327902205317211212502892905438050361", "1": "12.64143999522579871232479759208836567080484864417230303123038547612827769178465435019034486198960706", "2": "3.122833559750939383778936119807481286257377193822905995495939550550480733759900321449284823263304026"}, "-1": {"1/2": "4.144834046364355537462478018880087089605875138187282173476452058937544169049739380567093928295490506", "1": "1.630336124636594500754064590072031403373321233575084887052011483248774056054015004167334065491362195", "2": "0.7649019952684789830725280450568848692857054929443030478321510998474571991988689887091516428424062571"}, "2": {"1/2": "371.6188157914264509166222497518787692134478490697168907602547888358663734790114771142923644231635676", "1": "66.49640933288923121018738858732541300105771463352491576223107139745225326121515362781556560462895919", "2": "10.57839499360668752807611259868610492093339717699678715019209228113688905367266348284414072413721687", "5": "1.496877074998357979395992654859848387486288040027978361923528005603136699119914824317969327980856709"}, "-2": {"1/2": "1.982694432368566653244011849741715741923851846645770823432765627647029441239217504718545004142292122", "1": "1.011349998675990089550429065790597701980883564673626484317402932221123235326320743215442168292268910", "2": "0.09535043858249188227156721102317486410276964365301387684414083946809481842684330622445580617853086279"}}, "3": {"0": {"1/2": "123.0650234618082303566823990989323143119240799963477975598703539525292474505961814201940170875222318", "1": "16.64331454012380854974346196981097542830667663673849496844939923831664921444760124594252311438738555", "2": "2.968733114886159847711318889445655780917952038941686240141719502214870632376743812914196644636328579"}, "1/2": {"1/2": "257.2698165911684663718956610077634356542564736825559572244481598319924780171068794704563903512481509", "1": "31.90419127669482083664632972351754929629992355854619754037040386484236164094315944642723377616165660", "2": "4.646085364211713669521298474877940271907565111020896631818526169431024273020888437550530070139503233"}, "-1/2": {"1/2": "63.27623233058589742159457849066393562016179062513823184299661128033941906640522054679606033207768604", "1": "9.386137163525166950214553529271099021405440245936220185312003475353544208025308855212991299291005772", "2": "2.082187788299257754285499508094555997044952138630652044645461711775258583573541638379966497997507282"}, "1": {"1/2": "577.2463974206466276674065035352364873832304411445183005302468372947548582550392505784943192201601008", "1": "65.94520642806816588511738028797163086329248217119323699209075199252583891069103151845641157746220003", "2": "8.002969059737291685722982546159064077088246429727911734837338738427291698128887454363966629672506108"}, "-1": {"1/2": "34.92927216895851168757024751714634659240942028376874932772703736397952859876131630215848538127385541", "1": "5.722937656810725333608846923835988097745564881776942074556710749999204435092973005347926867733624985", "2": "1.576135692957533088517227206126027708628505954682807387534874128964131493095167563920889328827944837"}, "2": {"1/2": "3531.873090846641213034642084642524327322841473052760262960466430618510097585278697420065976587288053", "1": "346.1151958608465199925846030236134500975986224499993357362954909459665507643489024279838393311149878", "2": "30.70838509300982097635550697264024297823197048834771397982110727423017390076502935945709163137627288", "5": "2.048694713777195592942384636612149848890630178081008703408405290702015596033207036444701753170042215"}, "-2": {"1/2": "12.96258807172269364929819123357757633638809496752679447144540621296057972612041022918033414265073721", "1": "2.653802966566984975591161319560153938816743165302320567633103915490404486212929376174020235302124935", "2": "1.000715816794757147736827906209510760769444575356232259552789362254605285730450525228645244413160576"}}}, "Comments": {"comment-normalization": "This is the DLMF normalization with $H_\\ell^+=G_\\ell+iF_\\ell$, where $F_\\ell$ is the HREF{T510}[regular Coulomb wave function]. The asymptotic condition fixes both the sign of $G_\\ell$ and the possible addition of a multiple of $F_\\ell$. Values retain their signs.", "comment-physical": "Positive $\\eta$ describes repulsion and negative $\\eta$ attraction. In scattering, $\\rho=kr$, where $r$ is the separation and $k$ is the wave number CITE{DLMFPhysical}. The orders $\\ell=0,1,2,3$ are the s, p, d and f partial waves."}, "Formulas": {"formula-equation": "$\\frac{d^2w}{d\\rho^2}+\\left(1-\\frac{2\\eta}{\\rho}-\\frac{\\ell(\\ell+1)}{\\rho^2}\\right)w=0$ CITE{DLMFDefinition}.", "formula-phase": "$\\theta_\\ell(\\eta,\\rho)=\\rho-\\eta\\log(2\\rho)-\\ell\\pi/2+\\sigma_\\ell(\\eta)$, where $\\sigma_\\ell(\\eta)=\\operatorname{Im}\\log\\Gamma(\\ell+1+i\\eta)$ is continuous in real $\\eta$ and is zero at $\\eta=0$ CITE{DLMFDefinition}.", "formula-hypergeometric": "$G_\\ell(\\eta,\\rho)=\\operatorname{Re}\\!\\left[e^{i\\theta_\\ell(\\eta,\\rho)}(-2i\\rho)^{\\ell+1+i\\eta}U(\\ell+1+i\\eta,2\\ell+2,-2i\\rho)\\right]$, where $U$ is HREF{Values_of_Tricomi's_confluent_hypergeometric_function}[Tricomi's confluent hypergeometric function]. The power and $U$ use their principal branches; $\\theta_\\ell$ is defined in CITE{formula-phase} CITE{DLMFDefinition}.", "formula-free": "$G_\\ell(0,\\rho)=-\\sqrt{\\pi\\rho/2}\\,Y_{\\ell+1/2}(\\rho)$, where $Y_\\nu$ is HREF{Values_of_Bessel_functions_of_the_second_kind}[the Bessel function of the second kind]. In particular, $G_0(0,\\rho)=\\cos\\rho$ CITE{DLMFLimits}.", "formula-order": "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 CITE{DLMFRecurrence}.", "formula-derivative": "With $R_j,S_j$ from CITE{formula-order}, $\\partial_\\rho G_\\ell=S_{\\ell+1}G_\\ell-R_{\\ell+1}G_{\\ell+1}$ for $\\ell\\geq0$, and $\\partial_\\rho G_\\ell=R_\\ell G_{\\ell-1}-S_\\ell G_\\ell$ for $\\ell\\geq1$ CITE{DLMFRecurrence}.", "formula-wronskian": "$G_\\ell\\,\\partial_\\rho F_\\ell-F_\\ell\\,\\partial_\\rho G_\\ell=1$, where $F_\\ell$ is the HREF{T510}[regular Coulomb wave function] CITE{DLMFDefinition}."}, "Similar tables": [{"table": "HREF{T510}[Values of the regular Coulomb wave function]", "relation": "the companion solution $F_\\ell$ satisfies CITE{formula-wronskian} with $G_\\ell$"}, {"table": "HREF{Values_of_Bessel_functions_of_the_second_kind}[Values of Bessel functions of the second kind]", "relation": "half-integer orders give the free case $\\eta=0$ through CITE{formula-free}"}, {"table": "HREF{Values_of_Tricomi's_confluent_hypergeometric_function}[Values of Tricomi's confluent hypergeometric function]", "relation": "$U$ gives $G_\\ell$ through CITE{formula-hypergeometric}; the linked table holds real parameters and positive real arguments"}], "Keywords": ["CoulombG", "Coulomb wave", "irregular Coulomb wave function", "radial Coulomb", "CoulombF connection", "Coulomb scattering"], "Programs": {"program-python": {"language": "Python", "code": "import mpmath as mp\n\nmp.mp.dps = 150\n# One value at angular momentum 0, coupling 1, and radius 1.\nmp.nstr(mp.coulombg(0, 1, 1), 100)"}, "program-sage": {"language": "Sage", "code": "import numberdb.sage as numberdb\nfrom sage.rings.complex_arb import ComplexBallField\n\nC = ComplexBallField(numberdb.bits(150))\ni = C(0, 1)\nell, eta, rho = 0, C(1), C(1)\na, b, z = ell+1+i*eta, C(2*ell+2), -2*i*rho\ngamma = a.gamma()\nh = (-i)**ell * (C.pi()*eta/2).exp() * gamma/abs(gamma)\nh *= (i*rho).exp() * z**(ell+1) * z.hypergeometric_U(a, b)\nassert h.real().is_finite() and h.imag().is_finite()\nh.real()"}}}