# Fitting data set with series expansion

I have a function given in a form of a list:

ScalarField={{$\epsilon$,$y_1$},..{1,$y_{3000}$}}

It is defined in the range $(\epsilon,1)$ (spacing between grid points is homogeneous), where $\epsilon=10^{-5}$, and I know that as $z\rightarrow \epsilon$ it behaves as

$y(z)\sim C1\cdot z^5+C2\cdot z^7+B1\cdot z^{7.2}+C3\cdot z^{9}+B2\cdot z^{9.2}+...$

So, its series expansion contains integer valued powers $5+2k$ and fractional valued powers $7.2+2k$, $k=0...\infty$.

I need to know first three coefficients in this expansion - $C1,\,C2,\,B1$.

Standard fitting procedure can give me precise value for $C1$, reasonable $C2$, but $B1$ is highly fluctuating and strongly dependent on the number of points near the left boundary of the interval that I use for fit.

Does anybody know how $B1$ can be extracted?

Andrey

UPD!

Before coming to the real data set, which contains some extra numerical uncertainties, let's analyze simpler case.

The following list has asymptotic behaviour $A1\cdot z^5+A2\cdot z^7+A3\cdot z^9+...$ near zero. How to extract A3?

{{1/100000, -8.53187*10^-25}, {51499/ 149950000, -1.60222*10^-17}, {202997/ 299900000, -4.76459*10^-16}, {75749/ 74975000, -3.52989*10^-15}, {80599/ 59980000, -1.46917*10^-14}, {251497/ 149950000, -4.45028*10^-14}, {602993/ 299900000, -1.10188*10^-13}, {43937/ 18743750, -2.37313*10^-13}, {802991/ 299900000, -4.61446*10^-13}, {90299/ 29990000, -8.29812*10^-13}, {1002989/ 299900000, -1.40295*10^-12}, {275747/ 74975000, -2.25638*10^-12}, {1202987/ 299900000, -3.48225*10^-12}, {651493/ 149950000, -5.19099*10^-12}, {280597/ 59980000, -7.51297*10^-12}, {187873/ 37487500, -1.06002*10^-11}, {1602983/ 299900000, -1.46279*10^-11}, {851491/ 149950000, -1.97962*10^-11}, {1802981/ 299900000, -2.63319*10^-11}, {95149/ 14995000, -3.44898*10^-11}, {2002979/ 299900000, -4.45549*10^-11}, {1051489/ 149950000, -5.68435*10^-11}, {2202977/ 299900000, -7.17052*10^-11}, {71968/ 9371875, -8.95242*10^-11}, {96119/ 11996000, -1.10721*10^-10}, {1251487/ 149950000, -1.35756*10^-10}, {2602973/ 299900000, -1.65126*10^-10}, {675743/ 74975000, -1.99373*10^-10}, {2802971/ 299900000, -2.39079*10^-10}, {290297/ 29990000, -2.84873*10^-10}, {3002969/ 299900000, -3.37429*10^-10}, {387871/ 37487500, -3.9747*10^-10}, {3202967/ 299900000, -4.65768*10^-10}, {1651483/ 149950000, -5.43146*10^-10}, {680593/ 59980000, -6.30481*10^-10}, {875741/ 74975000, -7.28704*10^-10}, {3602963/ 299900000, -8.38804*10^-10}, {1851481/ 149950000, -9.61825*10^-10}, {3802961/ 299900000, -1.09887*10^-9}, {48787/ 3748750, -1.25111*10^-9}, {4002959/ 299900000, -1.41978*10^-9}, {2051479/ 149950000, -1.60615*10^-9}, {4202957/ 299900000, -1.81161*10^-9}, {1075739/ 74975000, -2.03757*10^-9}, {880591/ 59980000, -2.28553*10^-9}, {2251477/ 149950000, -2.55705*10^-9}, {4602953/ 299900000, -2.85378*10^-9}, {587869/ 37487500, -3.17744*10^-9}, {4802951/ 299900000, -3.5298*10^-9}, {98059/ 5998000, -3.91275*10^-9}, {5002949/ 299900000, -4.32822*10^-9}, {1275737/ 74975000, -4.77825*10^-9}, {5202947/ 299900000, -5.26495*10^-9}, {2651473/ 149950000, -5.79051*10^-9}, {1080589/ 59980000, -6.35722*10^-9}, {171967/ 9371875, -6.96743*10^-9}, {5602943/ 299900000, -7.62361*10^-9}, {2851471/ 149950000, -8.32831*10^-9}, {5802941/ 299900000, -9.08417*10^-9}, {295147/ 14995000, -9.89392*10^-9}, {6002939/ 299900000, -1.07604*10^-8}, {3051469/ 149950000, -1.16865*10^-8}, {6202937/ 299900000, -1.26753*10^-8}, {787867/ 37487500, -1.37299*10^-8}, {1280587/ 59980000, -1.48535*10^-8}, {3251467/ 149950000, -1.60495*10^-8}, {6602933/ 299900000, -1.73212*10^-8}, {1675733/ 74975000, -1.86723*10^-8}, {6802931/ 299900000, -2.01064*10^-8}, {690293/ 29990000, -2.16272*10^-8}, {7002929/ 299900000, -2.32386*10^-8}, {443933/ 18743750, -2.49446*10^-8}, {7202927/ 299900000, -2.67493*10^-8}, {3651463/ 149950000, -2.86569*10^-8}, {296117/ 11996000, -3.06718*10^-8}, {1875731/ 74975000, -3.27984*10^-8}, {7602923/ 299900000, -3.50412*10^-8}, {3851461/ 149950000, -3.7405*10^-8}, {7802921/ 299900000, -3.98947*10^-8}, {197573/ 7497500, -4.2515*10^-8}, {8002919/ 299900000, -4.52712*10^-8}, {4051459/ 149950000, -4.81685*10^-8}, {8202917/ 299900000, -5.12121*10^-8}, {2075729/ 74975000, -5.44075*10^-8}, {1680583/ 59980000, -5.77604*10^-8}, {4251457/ 149950000, -6.12764*10^-8}, {8602913/ 299900000, -6.49615*10^-8}, {271966/ 9371875, -6.88217*10^-8}, {8802911/ 299900000, -7.2863*10^-8}, {890291/ 29990000, -7.70918*10^-8}, {9002909/ 299900000, -8.15146*10^-8}, {2275727/ 74975000, -8.61378*10^-8}, {9202907/ 299900000, -9.09683*10^-8}, {4651453/ 149950000, -9.60129*10^-8}, {1880581/ 59980000, -1.01279*10^-7}, {1187863/ 37487500, -1.06773*10^-7}, {9602903/ 299900000, -1.12503*10^-7}, {4851451/ 149950000, -1.18475*10^-7}, {9802901/ 299900000, -1.24699*10^-7}, {99029/ 2999000, -1.31181*10^-7}, {10002899/ 299900000, -1.3793*10^-7}, {5051449/ 149950000, -1.44953*10^-7}, {10202897/ 299900000, -1.5226*10^-7}, {643931/ 18743750, -1.59857*10^-7}, {2080579/ 59980000, -1.67755*10^-7}, {5251447/ 149950000, -1.75962*10^-7}, {10602893/ 299900000, -1.84486*10^-7}, {2675723/ 74975000, -1.93337*10^-7}, {10802891/ 299900000, -2.02524*10^-7}, {1090289/ 29990000, -2.12056*10^-7}, {11002889/ 299900000, -2.21944*10^-7}, {1387861/ 37487500, -2.32196*10^-7}, {11202887/ 299900000, -2.42824*10^-7}, {5651443/ 149950000, -2.53836*10^-7}, {2280577/ 59980000, -2.65244*10^-7}, {2875721/ 74975000, -2.77058*10^-7}, {11602883/ 299900000, -2.89288*10^-7}, {5851441/ 149950000, -3.01946*10^-7}, {11802881/ 299900000, -3.15043*10^-7}, {74393/ 1874375, -3.28589*10^-7}, {12002879/ 299900000, -3.42597*10^-7}, {6051439/ 149950000, -3.57078*10^-7}, {12202877/ 299900000, -3.72044*10^-7}, {3075719/ 74975000, -3.87506*10^-7}, {99223/ 2399200, -4.03478*10^-7}, {6251437/ 149950000, -4.19971*10^-7}, {12602873/ 299900000, -4.36998*10^-7}, {1587859/ 37487500, -4.54572*10^-7}, {12802871/ 299900000, -4.72706*10^-7}, {1290287/ 29990000, -4.91413*10^-7}, {13002869/ 299900000, -5.10707*10^-7}, {3275717/ 74975000, -5.30601*10^-7}, {13202867/ 299900000, -5.51109*10^-7}, {6651433/ 149950000, -5.72246*10^-7}, {2680573/ 59980000, -5.94024*10^-7}, {843929/ 18743750, -6.16459*10^-7}, {13602863/ 299900000, -6.39566*10^-7}, {6851431/ 149950000, -6.63359*10^-7}, {13802861/ 299900000, -6.87854*10^-7}, {695143/ 14995000, -7.13065*10^-7}, {14002859/ 299900000, -7.39009*10^-7}, {7051429/ 149950000, -7.65701*10^-7}, {14202857/ 299900000, -7.93157*10^-7}, {1787857/ 37487500, -8.21393*10^-7}, {2880571/ 59980000, -8.50426*10^-7}, {7251427/ 149950000, -8.80272*10^-7}, {14602853/ 299900000, -9.10949*10^-7}, {3675713/ 74975000, -9.42473*10^-7}, {14802851/ 299900000, -9.74862*10^-7}, {298057/ 5998000, -1.00813*10^-6}, {15002849/ 299900000, -1.0423*10^-6}, {471964/ 9371875, -1.07739*10^-6}, {15202847/ 299900000, -1.11342*10^-6}, {7651423/ 149950000, -1.1504*10^-6}, {3080569/ 59980000, -1.18836*10^-6}, {3875711/ 74975000, -1.22731*10^-6}, {15602843/ 299900000, -1.26727*10^-6}, {7851421/ 149950000, -1.30826*10^-6}, {15802841/ 299900000, -1.3503*10^-6}, {397571/ 7497500, -1.39341*10^-6}, {16002839/ 299900000, -1.43762*10^-6}, {8051419/ 149950000, -1.48293*10^-6}, {16202837/ 299900000, -1.52938*10^-6}, {4075709/ 74975000, -1.57698*10^-6}, {3280567/ 59980000, -1.62576*10^-6}, {8251417/ 149950000, -1.67573*10^-6}, {16602833/ 299900000, -1.72692*10^-6}, {1043927/ 18743750, -1.77936*10^-6}, {16802831/ 299900000, -1.83305*10^-6}, {1690283/ 29990000, -1.88802*10^-6}, {17002829/ 299900000, -1.94431*10^-6}, {4275707/ 74975000, -2.00192*10^-6}, {17202827/ 299900000, -2.06089*10^-6}, {8651413/ 149950000, -2.12124*10^-6}, {696113/ 11996000, -2.18299*10^-6}, {2187853/ 37487500, -2.24616*10^-6}, {17602823/ 299900000, -2.31078*10^-6}, {8851411/ 149950000, -2.37687*10^-6}, {17802821/ 299900000, -2.44447*10^-6}, {895141/ 14995000, -2.51359*10^-6}, {18002819/ 299900000, -2.58425*10^-6}, {9051409/ 149950000, -2.6565*10^-6}, {18202817/ 299900000, -2.73034*10^-6}, {571963/ 9371875, -2.80581*10^-6}, {3680563/ 59980000, -2.88294*10^-6}, {9251407/ 149950000, -2.96174*10^-6}, {18602813/ 299900000, -3.04226*10^-6}, {4675703/ 74975000, -3.1245*10^-6}, {18802811/ 299900000, -3.20852*10^-6}, {1890281/ 29990000, -3.29432*10^-6}, {19002809/ 299900000, -3.38194*10^-6}, {2387851/ 37487500, -3.47141*10^-6}, {19202807/ 299900000, -3.56275*10^-6}, {9651403/ 149950000, -3.656*10^-6}, {3880561/ 59980000, -3.75119*10^-6}, {4875701/ 74975000, -3.84834*10^-6}, {19602803/ 299900000, -3.94748*10^-6}, {9851401/ 149950000, -4.04865*10^-6}, {19802801/ 299900000, -4.15187*10^-6}, {49757/ 749750, -4.25718*10^-6}, {20002799/ 299900000, -4.36461*10^-6}, {10051399/ 149950000, -4.47419*10^-6}, {20202797/ 299900000, -4.58595*10^-6}, {5075699/ 74975000, -4.69992*10^-6}, {4080559/ 59980000, -4.81613*10^-6}, {10251397/ 149950000, -4.93463*10^-6}, {20602793/ 299900000, -5.05543*10^-6}, {2587849/ 37487500, -5.17858*10^-6}, {20802791/ 299900000, -5.3041*10^-6}, {2090279/ 29990000, -5.43204*10^-6}, {21002789/ 299900000, -5.56242*10^-6}, {5275697/ 74975000, -5.69529*10^-6}, {21202787/ 299900000, -5.83067*10^-6}, {10651393/ 149950000, -5.96859*10^-6}, {4280557/ 59980000, -6.10911*10^-6}, {671962/ 9371875, -6.25224*10^-6}, {21602783/ 299900000, -6.39804*10^-6}, {10851391/ 149950000, -6.54652*10^-6}, {21802781/ 299900000, -6.69774*10^-6}, {1095139/ 14995000, -6.85172*10^-6}, {22002779/ 299900000, -7.00851*10^-6}, {11051389/ 149950000, -7.16815*10^-6}, {22202777/ 299900000, -7.33066*10^-6}, {2787847/ 37487500, -7.49609*10^-6}, {896111/ 11996000, -7.66448*10^-6}, {11251387/ 149950000, -7.83587*10^-6}, {22602773/ 299900000, -8.01029*10^-6}, {5675693/ 74975000, -8.18779*10^-6}, {22802771/ 299900000, -8.3684*10^-6}, {2290277/ 29990000, -8.55217*10^-6}, {23002769/ 299900000, -8.73914*10^-6}, {1443923/ 18743750, -8.92935*10^-6}, {23202767/ 299900000, -9.12283*10^-6}, {11651383/ 149950000, -9.31964*10^-6}, {4680553/ 59980000, -9.5198*10^-6}, {5875691/ 74975000, -9.72338*10^-6}, {23602763/ 299900000, -9.9304*10^-6}, {11851381/ 149950000, -0.0000101409}, {23802761/ 299900000, -0.000010355}, {597569/ 7497500, -0.0000105726}, {24002759/ 299900000, -0.0000107938}, {12051379/ 149950000, -0.0000110187}, {24202757/ 299900000, -0.0000112474}, {6075689/ 74975000, -0.0000114798}, {4880551/ 59980000, -0.0000117159}, {12251377/ 149950000, -0.000011956}, {24602753/ 299900000, -0.0000121999}, {771961/ 9371875, -0.0000124478}, {24802751/ 299900000, -0.0000126996}, {99611/ 1199600, -0.0000129555}, {25002749/ 299900000, -0.0000132155}, {6275687/ 74975000, -0.0000134795}, {25202747/ 299900000, -0.0000137478}, {12651373/ 149950000, -0.0000140203}, {5080549/ 59980000, -0.0000142971}, {3187843/ 37487500, -0.0000145782}, {25602743/ 299900000, -0.0000148637}, {12851371/ 149950000, -0.0000151536}, {25802741/ 299900000, -0.0000154479}, {1295137/ 14995000, -0.0000157468}, {26002739/ 299900000, -0.0000160503}, {13051369/ 149950000, -0.0000163584}, {26202737/ 299900000, -0.0000166712}, {1643921/ 18743750, -0.0000169887}, {5280547/ 59980000, -0.000017311}, {13251367/ 149950000, -0.0000176382}, {26602733/ 299900000, -0.0000179702}, {6675683/ 74975000, -0.0000183072}, {26802731/ 299900000, -0.0000186491}, {2690273/ 29990000, -0.0000189962}, {27002729/ 299900000, -0.0000193483}, {3387841/ 37487500, -0.0000197056}, {27202727/ 299900000, -0.0000200681}, {13651363/ 149950000, -0.0000204359}, {1096109/ 11996000, -0.000020809}, {6875681/ 74975000, -0.0000211876}, {27602723/ 299900000, -0.0000215715}, {13851361/ 149950000, -0.000021961}, {27802721/ 299900000, -0.000022356}, {174392/ 1874375, -0.0000227566}, {28002719/ 299900000, -0.000023163}, {14051359/ 149950000, -0.000023575}, {28202717/ 299900000, -0.0000239929}, {7075679/ 74975000, -0.0000244166}, {5680543/ 59980000, -0.0000248462}, {14251357/ 149950000, -0.0000252818}, {28602713/ 299900000, -0.0000257234}, {3587839/ 37487500, -0.0000261712}, {28802711/ 299900000, -0.0000266251}, {2890271/ 29990000, -0.0000270852}, {29002709/ 299900000, -0.0000275516}, {7275677/ 74975000, -0.0000280243}, {29202707/ 299900000, -0.0000285035}, {14651353/ 149950000, -0.0000289891}, {5880541/ 59980000, -0.0000294813}, {1843919/ 18743750, -0.0000299801}, {29602703/ 299900000, -0.0000304855}, {14851351/ 149950000, -0.0000309977}, {29802701/ 299900000, -0.0000315167}, {299027/2999000, -0.0000320425}}

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Are the first 300 points representative of the very long list? – Peltio Dec 7 '13 at 1:32
The plot of y[z] / z^5 near 0behaves like 1 /z^n + 0.154 for some n, and not like z^2. Are you sure your expansion is OK? – Dr. belisarius Dec 7 '13 at 1:43
Belisarius, thanks for your comment. Sure, very close to zero the numerical data is not very reliable, that's another problem. So, first 40-50 point should be thrown away. y[z]/z^5 ideally should behave like a constant. – user10998 Dec 7 '13 at 2:22
Peltio, yes, here is first 300 of 3000 points. – user10998 Dec 7 '13 at 2:23
Problem is reformulated. Let's solve its simpler version first. – user10998 Dec 7 '13 at 2:28