Given an analytic function of a complex number in terms of a Taylor series, analytic continuation is a technique for evaluation of the function beyond the original radius of convergence, which is often finite. You choose another point within that circle, presumably further from the singularity on the circle responsible for the first limit. You formally differentiate the first series to get the coefficients of the second series.

This routine does just that and here is a slightly uglier version that reports the number of terms required for a particular evaluation.

Logic

The function we consider here is the dilogarithm, li2(z) = Li₂(z) = Σ[k>0]z^k/k². There is a singularity at z=1 and only the unit circle is covered by this definition. Formally differentiating Li₂ provides a series:
dli2(n, z)*n! = dⁿLi₂(z)/(dz)ⁿ = Σ[k >= n](k!/(k-n)!)z^k-n/k². The array u stores the coefficients of the new series. Those are real if the new point, nc is real. li2h(z) evaluates Li₂ with the new Taylor series.

Here is the yield of the program:

c0=        +48
+8.109302162163287642884046469e  -1+0.000000000000000000000000000e  +0        +54
+1.442635495496620978580698474e  -1+0.000000000000000000000000000e  +0        +58
+4.420325125140131452066425552e  -2+0.000000000000000000000000000e  +0        +63
+1.692216082771925586319314735e  -2+0.000000000000000000000000000e  +0        +67
+7.322370904597723207600811096e  -3+0.000000000000000000000000000e  +0        +71
+3.424801987772611703442886044e  -3+0.000000000000000000000000000e  +0        +75
+1.690541731471972500610353859e  -3+0.000000000000000000000000000e  +0        +79
+8.681743348639854188042635870e  -4+0.000000000000000000000000000e  +0        +83
+4.595754200186591992024880612e  -4+0.000000000000000000000000000e  +0        +86
+2.491847588391655356226690149e  -4+0.000000000000000000000000000e  +0        +90
+1.377626539651620526530030260e  -4+0.000000000000000000000000000e  +0        +94
+7.739789675600168823422917859e  -5+0.000000000000000000000000000e  +0        +97
+4.407628850776092728226075603e  -5+0.000000000000000000000000000e  +0       +101
+2.539188650152765066617754849e  -5+0.000000000000000000000000000e  +0       +105
+1.477449870016947319493230358e  -5+0.000000000000000000000000000e  +0       +108
+8.671699558474135762264376415e  -6+0.000000000000000000000000000e  +0       +112
+5.128795931042352586351040026e  -6+0.000000000000000000000000000e  +0       +115
+3.054005210281280704382936902e  -6+0.000000000000000000000000000e  +0       +119
+1.829580137616931634311185753e  -6+0.000000000000000000000000000e  +0       +122
+1.102028631300093924263241111e  -6+0.000000000000000000000000000e  +0       +126
+6.670609176097356485955426283e  -7+0.000000000000000000000000000e  +0       +129
+4.055761534699509593827476483e  -7+0.000000000000000000000000000e  +0       +133
+2.475955578529450562263401151e  -7+0.000000000000000000000000000e  +0       +136
+1.517147174228864304491089583e  -7+0.000000000000000000000000000e  +0       +140
+9.328161723888008873856241786e  -8+0.000000000000000000000000000e  +0       +143
+5.753502379214177514479244534e  -8+0.000000000000000000000000000e  +0       +146
+3.559047688978763233060494652e  -8+0.000000000000000000000000000e  +0       +150
+2.207542818912257955297288515e  -8+0.000000000000000000000000000e  +0       +153
+1.372702422697430283414069932e  -8+0.000000000000000000000000000e  +0       +157
+8.555827124632215069875135569e  -9+0.000000000000000000000000000e  +0       +160
+5.344407227999234305903214594e  -9+0.000000000000000000000000000e  +0       +164
+3.345252644311093417025050988e  -9+0.000000000000000000000000000e  +0       +167
+2.097952056846604968808976097e  -9+0.000000000000000000000000000e  +0       +170
+1.318104173059400969921175012e  -9+0.000000000000000000000000000e  +0       +174
+8.295566234068796578451788282e -10+0.000000000000000000000000000e  +0       +177
+5.229291681385937826614714487e -10+0.000000000000000000000000000e  +0       +181
+3.301429589616324966919022049e -10+0.000000000000000000000000000e  +0       +184
+2.087316960203159477295030005e -10+0.000000000000000000000000000e  +0       +188
+1.321507190041562557354436824e -10+0.000000000000000000000000000e  +0       +191
+8.377519807200752652268134966e -11+0.000000000000000000000000000e  +0       +194
+5.317388963893066451108737306e -11+0.000000000000000000000000000e  +0       +198
+3.379034779726484284761555217e -11+0.000000000000000000000000000e  +0       +201
+2.149682139713669169827282810e -11+0.000000000000000000000000000e  +0       +204
+1.369054812455794026046041835e -11+0.000000000000000000000000000e  +0       +208
+8.727936792371253160989850205e -12+0.000000000000000000000000000e  +0       +211
+5.569640399589331236140302949e -12+0.000000000000000000000000000e  +0       +215
+3.557536802249499564002882732e -12+0.000000000000000000000000000e  +0       +218
+2.274369008127087545172805490e -12+0.000000000000000000000000000e  +0       +221
+1.455278641234912730349658855e -12+0.000000000000000000000000000e  +0       +225
+9.319440444469279433986432223e -13+0.000000000000000000000000000e  +0       +228
+5.972799123351414090394701764e -13+0.000000000000000000000000000e  +0       +232
+3.830836295251898503586992161e -13+0.000000000000000000000000000e  +0       +235
+2.458889533515047352991095554e -13+0.000000000000000000000000000e  +0       +238
+1.579328305613038146535592057e -13+0.000000000000000000000000000e  +0       +242
+1.015145482233771667365455387e -13+0.000000000000000000000000000e  +0       +245
+6.529969072659700123808053975e -14+0.000000000000000000000000000e  +0       +249
+4.200719803114158934086993208e -14+0.000000000000000000000000000e  +0       +252
+2.710522557046082830395422792e -14+0.000000000000000000000000000e  +0       +255
+1.739196513247754779906860119e -14+0.000000000000000000000000000e  +0       +259
+1.101656364840764932524429690e -14+0.000000000000000000000000000e  +0       +262
+6.648718299834426829997196968e -15+0.000000000000000000000000000e  +0       +265
+6.054651900280824289401979588e -15+0.000000000000000000000000000e  +0       +269
+4.218955672123659249474089418e -15+0.000000000000000000000000000e  +0       +272
+5.728350038868643459851041532e -15+0.000000000000000000000000000e  +0       +276
+1.340709706650134857627827423e -15+0.000000000000000000000000000e  +0       +279
-1.086925381623007232226340199e -14+0.000000000000000000000000000e  +0       +282
+3.575315425538331599269625666e -14+0.000000000000000000000000000e  +0       +286
-2.666588028970306367232162391e -14+0.000000000000000000000000000e  +0       +289
-1.020645593362159256378929002e -13+0.000000000000000000000000000e  +0       +292
+7.501501707945035734661442385e -14+0.000000000000000000000000000e  +0       +296
+0.000000000000000000000000000e  +0+0.000000000000000000000000000e  +0         +0 z
-4.272083023905974289938489729e -19+0.000000000000000000000000000e  +0        +31 li2h
+0.000000000000000000000000000e  +0+0.000000000000000000000000000e  +0         -2 li2

-5.000000000000000000000000000e  -1+0.000000000000000000000000000e  +0         +0 z
-4.484142069236462019431971496e  -1+0.000000000000000000000000000e  +0         +0 li2h
-4.484142069236462019431971496e  -1+0.000000000000000000000000000e  +0        +48 li2

+1.000000000000000000000000000e  -1+0.000000000000000000000000000e  +0         +0 z
+1.026177910993911305604508079e  -1+0.000000000000000000000000000e  +0        +37 li2h
+1.026177910993911306852193699e  -1+0.000000000000000000000000000e  +0        +15 li2

-9.900000000000000000000000000e  -1+0.000000000000000000000000000e  +0         +0 z
-8.155258814773397489059956698e  -1+0.000000000000000000000000000e  +0        +31 li2h
-8.155258814773397485970393407e  -1+0.000000000000000000000000000e  +0      +2562 li2

-5.200000000000000000000000000e  -1+4.520000000000000000000000000e  -1         +0 z
-4.928647603129084981299903427e  -1+3.601212550290216021132153451e  -1        +29 li2h
-4.928647603129084989484401814e  -1+3.601212550290216014992831563e  -1        +87 li2

-1.000000000000000000000000000e  +0+0.000000000000000000000000000e  +0         +0 z
-8.224670334241132183248764639e  -1+0.000000000000000000000000000e  +0        +31 li2h
+0.000000000000000000000000000e  +0+0.000000000000000000000000000e  +0         -1 li2

+5.403023058681397174009366074e  -1+8.414709848078965066525023216e  -1         +0 z
22      FOR n FROM 0 WHILE C t = f*u[n]; ABS t > 1.e-18 DO s +:= t; f *:= z; d := n OD; 
                                     1                                                  
.1   (s, d));
a68g: runtime error: 1: index out of bounds (detected in VOID loop-clause starting at 
"FOR" in this line).

The coefficients of the new series are reported. Curiously they get smaller for a while and then tend to increase. I think that this is real. There is good agreement between the two series. The last test case is a point on the original convergence radius. It ran out of tabulated coefficients!