-
Notifications
You must be signed in to change notification settings - Fork 0
/
8.6 Gamma functions.nb
7839 lines (7597 loc) · 379 KB
/
8.6 Gamma functions.nb
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(* Content-type: application/mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 7.0' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 145, 7]
NotebookDataLength[ 379972, 7830]
NotebookOptionsPosition[ 367480, 7508]
NotebookOutlinePosition[ 369055, 7560]
CellTagsIndexPosition[ 369012, 7557]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[Cell[TextData[StyleBox["Rules for integrands involving gamma \
functions",
FontFamily->"Arial"]], "None"]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.4965234353062396`*^9, {3.5193214825316973`*^9, 3.5193214852617016`*^9}, {
3.523316399894187*^9, 3.5233164032341914`*^9}, {3.523318350558202*^9,
3.5233183516382036`*^9}, {3.536542725780803*^9, 3.536542725780803*^9}, {
3.5367764878952584`*^9, 3.5367764886452594`*^9}, {3.544322815209103*^9,
3.5443228192963104`*^9}, {3.5450662624913826`*^9, 3.545066292141424*^9}, {
3.545066402271579*^9, 3.5450664054415827`*^9}, {3.5450834884295015`*^9,
3.5450834895895033`*^9}, {3.5454961263762493`*^9,
3.5454961321462574`*^9}, {3.5454963863166127`*^9,
3.5454964580867133`*^9}, {3.545497046127537*^9, 3.545497046127537*^9}, {
3.5454971130676303`*^9, 3.545497136257663*^9}, 3.545610399977621*^9, {
3.546040778592962*^9, 3.546040778592962*^9}, {3.5461052065706367`*^9,
3.5461052249474688`*^9}, {3.546191339879622*^9, 3.5461913430596266`*^9}, {
3.5462141824067917`*^9, 3.5462142145768366`*^9}, {3.5462145455473003`*^9,
3.5462145601073203`*^9}, {3.5463149785974817`*^9,
3.5463149794195285`*^9}, {3.5488738437683954`*^9,
3.5488738699084315`*^9}, {3.5488890647645535`*^9, 3.548889080804576*^9}, {
3.5505951583099127`*^9, 3.5505951620351257`*^9}, {3.5758457803910656`*^9,
3.5758457938510847`*^9}, {3.5758458780912023`*^9,
3.5758458797312045`*^9}, {3.5758460250714083`*^9, 3.575846028691413*^9}, {
3.5758461031815176`*^9, 3.575846103681518*^9}, {3.5758463018517957`*^9,
3.5758463042917986`*^9}, {3.5758464155019546`*^9, 3.575846417191957*^9}, {
3.5758468610025787`*^9, 3.5758468696025906`*^9}, {3.5758469815427475`*^9,
3.5758469940427647`*^9}, {3.575847092102902*^9, 3.575847093372904*^9}, {
3.5758472070230627`*^9, 3.5758472111630683`*^9}},
TextAlignment->Center,
FontWeight->"Bold"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{Cell[TextData[StyleBox["1.",
FontFamily->"Arial"]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{"u", " ",
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9,
3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9,
3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9,
3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9,
3.5291962665099535`*^9}, {3.5291997050223927`*^9,
3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9,
3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9,
3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9,
3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9,
3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9,
3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9,
3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9,
3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9,
3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9},
3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9},
3.574964614077545*^9},
FontSize->12,
FontWeight->"Bold"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\t",
RowBox[{Cell[TextData[StyleBox["1:",
FontFamily->"Arial",
FontColor->RGBColor[1, 0, 0]]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9},
3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9,
3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9,
3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9,
3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9,
3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9},
3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9,
3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9,
3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9,
3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9,
3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9,
3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9,
3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9,
3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9},
3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9,
3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9},
3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9,
3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9},
3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9},
3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9,
3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9,
3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9,
3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9},
3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9},
3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599651210365`*^9,
3.57495996777104*^9}, {3.574961504633192*^9, 3.574961504973192*^9},
3.57496462482756*^9},
FontSize->12,
FontWeight->"Bold"],
Cell["Derivation: Integration by parts", "Subsubsection",
CellChangeTimes->{3.4941724567488375`*^9}],
Cell[TextData[{
"Basis: ",
Cell[BoxData[
RowBox[{
RowBox[{
SubscriptBox["\[PartialD]", "x"],
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}]}], "\[Equal]",
RowBox[{"-",
FractionBox[
RowBox[{"b", " ",
SuperscriptBox[
RowBox[{"(",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}], ")"}],
RowBox[{"n", "-", "1"}]]}],
SuperscriptBox["\[ExponentialE]",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]]]}]}]],
CellChangeTimes->{{3.721077959964218*^9, 3.72107796559554*^9}, {
3.7210781170462027`*^9, 3.721078164012889*^9}}]
}], "Subsubsection",
CellChangeTimes->{
3.4941724567488375`*^9, {3.7210781295989203`*^9, 3.72107813133502*^9}, {
3.72107817365144*^9, 3.7210781742474747`*^9}}],
Cell["Rule:", "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,
3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9,
3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9,
3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9,
3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9},
3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9},
3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439179913333*^9}],
Cell[BoxData[
RowBox[{
RowBox[{"\[Integral]",
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}],
RowBox[{"\[DifferentialD]",
RowBox[{"x", " ", "\[LongRightArrow]", " ",
FractionBox[
RowBox[{
RowBox[{"(",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}], ")"}], " ",
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}]}], "b"]}]}]}]}], "+",
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
SuperscriptBox[
RowBox[{"(",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}], ")"}], "n"],
SuperscriptBox["\[ExponentialE]",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]]],
RowBox[{"\[DifferentialD]",
RowBox[{"x", " ", " ", "\[LongRightArrow]", " ",
FractionBox[
RowBox[{
RowBox[{"(",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}], ")"}], " ",
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}]}], "b"]}]}]}]}], "-",
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{
RowBox[{"n", "+", "1"}], ",",
RowBox[{"a", "+",
RowBox[{"b", " ", "x"}]}]}], "]"}], "b"]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9,
3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9,
3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9,
3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9,
3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9},
3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9,
3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9},
3.4941724567488375`*^9, 3.4944391698033185`*^9, 3.4944392275633993`*^9, {
3.7210780475842295`*^9, 3.7210780582708406`*^9}, {3.7210781941136103`*^9,
3.7210782613824577`*^9}, 3.721078305663991*^9},
TextAlignment->Center,
FontSize->12,
FontWeight->"Bold"],
Cell["Program code:", "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,
3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9,
3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9,
3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9,
3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9},
3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9,
3.492805165713872*^9}, 3.4941724567488375`*^9}],
Cell[BoxData[
RowBox[{
RowBox[{"Int", "[",
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"n_", ",",
RowBox[{"a_.", "+",
RowBox[{"b_.", "*", "x_"}]}]}], "]"}], ",", "x_Symbol"}], "]"}], " ", ":=",
"\n", " ",
RowBox[{
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"a", "+",
RowBox[{"b", "*", "x"}]}], ")"}], "*",
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"a", "+",
RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}]}], " ", "-", " ",
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{
RowBox[{"n", "+", "1"}], ",",
RowBox[{"a", "+",
RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}]}], " ", "/;", "\n",
RowBox[{"FreeQ", "[",
RowBox[{
RowBox[{"{",
RowBox[{"a", ",", "b", ",", "n"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9,
3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, {
3.721096929542216*^9, 3.7210969343364897`*^9}},
Background->GrayLevel[0.85]],
Cell["", "Subsubsection",
CellDingbat->None,
CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\t",
RowBox[{Cell[TextData[StyleBox["2.",
FontFamily->"Arial"]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"d", " ", "x"}], ")"}], "m"], " ",
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"b", " ", "x"}]}], "]"}],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9,
3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9,
3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9,
3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9,
3.5291962665099535`*^9}, {3.5291997050223927`*^9,
3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9,
3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9,
3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9,
3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9,
3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9,
3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9,
3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9,
3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9,
3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9},
3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9},
3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9},
3.574971287956888*^9, {3.574971361966992*^9, 3.574971373617008*^9},
3.709308508646453*^9, {3.7210769791321173`*^9, 3.7210769801441755`*^9}},
FontSize->12,
FontWeight->"Bold"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\t\t",
RowBox[{Cell[TextData[StyleBox["1.",
FontFamily->"Arial"]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9,
3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9,
3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9,
3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9,
3.5291962665099535`*^9}, {3.5291997050223927`*^9,
3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9,
3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9,
3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9,
3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9,
3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9,
3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9,
3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9,
3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9,
3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9},
3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9},
3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9},
3.574971287956888*^9, {3.574971361966992*^9, 3.574971373617008*^9},
3.7092687343291826`*^9, 3.709308499522931*^9, 3.7210769828293295`*^9},
FontSize->12,
FontWeight->"Bold"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\t\t\t",
RowBox[{Cell[TextData[StyleBox["1.",
FontFamily->"Arial"]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{"n", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]", "x"}], " ",
StyleBox["when",
FontFamily->"Arial",
FontWeight->"Plain"],
StyleBox[" ",
FontFamily->"Arial",
FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
RowBox[{"n", "\[Element]", "\[DoubleStruckCapitalZ]"}]]]],
"None"]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9,
3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9,
3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9,
3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9,
3.5291962665099535`*^9}, {3.5291997050223927`*^9,
3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9,
3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9,
3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9,
3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9,
3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9,
3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9,
3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9,
3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9,
3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9},
3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9},
3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9},
3.574971287956888*^9, {3.574971361966992*^9, 3.574971373617008*^9},
3.7092687343291826`*^9, 3.709308499522931*^9, {3.7093096690149946`*^9,
3.7093096776574097`*^9}, 3.7210769838043847`*^9},
FontSize->12,
FontWeight->"Bold"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\t\t\t\t",
RowBox[{Cell[TextData[StyleBox["1:",
FontFamily->"Arial",
FontColor->RGBColor[1, 0, 0]]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{"0", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9},
3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9,
3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9,
3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9,
3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9,
3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9},
3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9,
3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9,
3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9,
3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9,
3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9,
3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9,
3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9,
3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9},
3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9,
3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9},
3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9,
3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9},
3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9},
3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9,
3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9,
3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9,
3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9},
3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9},
3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9,
3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9,
3.5749646751076307`*^9}, {3.5749647914277935`*^9,
3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9,
3.7092687851164837`*^9}, 3.7093097077518625`*^9, 3.721076984466423*^9},
FontSize->12,
FontWeight->"Bold"],
Cell[TextData[{
"Basis: ",
Cell[BoxData[
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"0", ",", "z"}], "]"}], "\[Equal]",
RowBox[{"ExpIntegralE", "[",
RowBox[{"1", ",", "z"}], "]"}]}]],
CellDingbat->None,
CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9,
3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9,
3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9,
3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9,
3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9},
3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9,
3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9},
3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9},
3.4944395044737873`*^9, {3.5525056981388483`*^9,
3.5525057072180643`*^9}, {3.552505897366798*^9, 3.552505924198845*^9}, {
3.5525060501222663`*^9, 3.5525060654882936`*^9}, {3.5525061289648046`*^9,
3.5525061418972273`*^9}, {3.5525062448730087`*^9,
3.5525062459650106`*^9}, {3.5525100606611214`*^9,
3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9,
3.709268822873951*^9, 3.7092690350265255`*^9, {3.7092691334905024`*^9,
3.7092691676751623`*^9}},
TextAlignment->Center,
FontSize->12,
FontWeight->"Bold"]
}], "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,
3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9,
3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9,
3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9,
3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9},
3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9},
3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9,
3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9,
3.5525065695719786`*^9}, {3.5749853271085987`*^9, 3.5749853271085987`*^9},
3.7092691223832827`*^9, {3.7210767172471385`*^9, 3.721076734526127*^9}, {
3.721076771490241*^9, 3.721076771490241*^9}}],
Cell["Rule:", "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,
3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9,
3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9,
3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9,
3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9},
3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9},
3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9,
3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9,
3.5525065695719786`*^9}, {3.5749853271085987`*^9, 3.5749853271085987`*^9},
3.7092691223832827`*^9}],
Cell[BoxData[{
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{"0", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]",
RowBox[{"x", " ", "\[LongRightArrow]", " ",
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"ExpIntegralE", "[",
RowBox[{"1", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}], "\[IndentingNewLine]",
RowBox[{" ",
RowBox[{
RowBox[{
RowBox[{"\[LongRightArrow]", " ", "b"}], " ", "x", " ",
RowBox[{"HypergeometricPFQ", "[",
RowBox[{
RowBox[{"{",
RowBox[{"1", ",", "1", ",", "1"}], "}"}], ",",
RowBox[{"{",
RowBox[{"2", ",", "2", ",", "2"}], "}"}], ",",
RowBox[{
RowBox[{"-", "b"}], " ", "x"}]}], "]"}]}], "-",
RowBox[{"EulerGamma", " ",
RowBox[{"Log", "[", "x", "]"}]}], "-",
RowBox[{
FractionBox["1", "2"], " ",
SuperscriptBox[
RowBox[{"Log", "[",
RowBox[{"b", " ", "x"}], "]"}], "2"]}]}]}]}], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9,
3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9,
3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9,
3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9,
3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9},
3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9,
3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9},
3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9},
3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9,
3.5525060654882936`*^9}, {3.5525061289648046`*^9,
3.5525061418972273`*^9}, {3.5525062448730087`*^9,
3.5525062459650106`*^9}, {3.5525100606611214`*^9,
3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9,
3.709268822873951*^9, 3.7092690350265255`*^9, {3.7092691334905024`*^9,
3.7092691676751623`*^9}},
TextAlignment->Center,
FontSize->12,
FontWeight->"Bold"],
Cell["Program code:", "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,
3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9,
3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9,
3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9,
3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9},
3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9,
3.492805165713872*^9}, 3.4941724567488375`*^9}],
Cell[BoxData[
RowBox[{
RowBox[{"Int", "[",
RowBox[{
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"0", ",",
RowBox[{"b_.", "*", "x_"}]}], "]"}], "/", "x_"}], ",", "x_Symbol"}],
"]"}], " ", ":=", "\n", " ",
RowBox[{
RowBox[{
RowBox[{"b", "*", "x", "*",
RowBox[{"HypergeometricPFQ", "[",
RowBox[{
RowBox[{"{",
RowBox[{"1", ",", "1", ",", "1"}], "}"}], ",",
RowBox[{"{",
RowBox[{"2", ",", "2", ",", "2"}], "}"}], ",",
RowBox[{
RowBox[{"-", "b"}], "*", "x"}]}], "]"}]}], " ", "-", " ",
RowBox[{"EulerGamma", "*",
RowBox[{"Log", "[", "x", "]"}]}], " ", "-", " ",
RowBox[{
RowBox[{"1", "/", "2"}], "*",
RowBox[{
RowBox[{"Log", "[",
RowBox[{"b", "*", "x"}], "]"}], "^", "2"}]}]}], " ", "/;", "\n",
RowBox[{"FreeQ", "[",
RowBox[{"b", ",", "x"}], "]"}]}]}]], "Code",
CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9,
3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9,
3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
3.5749849658780932`*^9, 3.5749849692580976`*^9}, 3.7092692026232243`*^9,
3.7092692984867935`*^9, {3.709269335120658*^9, 3.7092693386774645`*^9},
3.7093099799381447`*^9},
Background->GrayLevel[0.85]],
Cell["", "Subsubsection",
CellDingbat->None,
CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\t\t\t\t",
RowBox[{Cell[TextData[StyleBox["x:",
FontFamily->"Arial",
FontColor->RGBColor[1, 0, 0]]], "None"], " ",
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{"1", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{
3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9,
3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9,
3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9},
3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9},
3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9,
3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9,
3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9,
3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9},
3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9},
3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9,
3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9,
3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9,
3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9,
3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9},
3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9,
3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9,
3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9,
3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9,
3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9,
3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9,
3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9,
3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9},
3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9,
3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9},
3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9,
3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9},
3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9},
3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9,
3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9,
3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9,
3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9},
3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9},
3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9,
3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9,
3.5749646751076307`*^9}, {3.5749647914277935`*^9,
3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9,
3.709268751309224*^9}, {3.709269573021481*^9, 3.7092696283899784`*^9}, {
3.70930932164629*^9, 3.70930932164629*^9}, {3.709309641442289*^9,
3.709309642060324*^9}, {3.709309711308669*^9, 3.7093097369933147`*^9}, {
3.709309836585691*^9, 3.709309863294938*^9}, {3.709314505135638*^9,
3.7093145123896513`*^9}, {3.709315130402023*^9, 3.709315130402023*^9}, {
3.709315482388446*^9, 3.7093154865108533`*^9}, {3.709315831200208*^9,
3.7093158320122547`*^9}, 3.72107698494745*^9},
FontSize->12,
FontWeight->"Bold"],
Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
CellChangeTimes->{
3.4941724567488375`*^9, {3.709313827569145*^9, 3.7093138311883516`*^9}}],
Cell[TextData[{
"Basis: ",
Cell[BoxData[
RowBox[{
RowBox[{"Gamma", "[",
RowBox[{"1", ",", "z"}], "]"}], "\[Equal]",
FractionBox["1",
SuperscriptBox["\[ExponentialE]", "z"]]}]], "Input",
CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}]
}], "Subsubsection",
CellChangeTimes->{
3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
3.709315648373751*^9, 3.709315648373751*^9}, {3.7210785613906174`*^9,
3.7210785632457237`*^9}}],
Cell[TextData[{
"Note: ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" automatically evaluates ",
Cell[BoxData[
RowBox[{"Gamma", "[",
RowBox[{"1", ",", "z"}], "]"}]],
CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}],
" to ",
Cell[BoxData[
SuperscriptBox["\[ExponentialE]",
RowBox[{"-", "z"}]]],
CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}],
"."
}], "Subsubsection",
CellChangeTimes->{
3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
3.709315648373751*^9, 3.709315648373751*^9}, {3.7093156864559293`*^9,
3.709315761858242*^9}, {3.721078568914048*^9, 3.721078593625461*^9}}],
Cell[TextData[{
"Rule: If ",
Cell[BoxData[
RowBox[{"n", ">", "1"}]]],
", then"
}], "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,
3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9,
3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9,
3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9,
3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9},
3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9},
3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9,
3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9,
3.5525065695719786`*^9}, {3.5749853271085987`*^9,
3.5749853271085987`*^9}, {3.7092696407919316`*^9,
3.7092696407919316`*^9}, {3.7093096482556787`*^9,
3.7093096482556787`*^9}, {3.7093097441793275`*^9,
3.7093097441793275`*^9}, {3.709309827146674*^9, 3.709309827146674*^9}, {
3.7093145217506676`*^9, 3.7093145217506676`*^9}, {3.7093151539554653`*^9,
3.7093151539554653`*^9}}],
Cell[BoxData[
RowBox[{"\[Integral]",
RowBox[{
FractionBox[
RowBox[{"Gamma", "[",
RowBox[{"1", ",",
RowBox[{"b", " ", "x"}]}], "]"}], "x"],
RowBox[{"\[DifferentialD]",
RowBox[{"x", " ", "\[LongRightArrow]", " ",
RowBox[{"\[Integral]",
RowBox[{
FractionBox["1",
RowBox[{"x", " ",
SuperscriptBox["\[ExponentialE]",
RowBox[{"b", " ", "x"}]]}]],
RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
CellDingbat->None,
CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9,
3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9,
3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9,
3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9,
3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9},
3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9,
3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9},
3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9},
3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9,
3.5525060654882936`*^9}, {3.5525061289648046`*^9,
3.5525061418972273`*^9}, {3.5525062448730087`*^9,
3.5525062459650106`*^9}, {3.5525100606611214`*^9,
3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9,
3.709309376380128*^9, 3.709309407253894*^9, {3.7093094939968557`*^9,
3.7093095366912975`*^9}, {3.709314182883684*^9, 3.7093142012777367`*^9}, {
3.709314257078928*^9, 3.7093143190864744`*^9}, {3.709315528139693*^9,
3.7093155364857073`*^9}, 3.709315658726343*^9},
TextAlignment->Center,
FontSize->12,
FontWeight->"Bold"],
Cell["Program code:", "Subsubsection",
CellDingbat->"\[FilledSmallSquare]",
CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9,