Hey, Ember!
I was just double checking everything before I uploaded it. It's about as good as I could get it. Here it is!
beta.gp (Size: 12.77 KB / Downloads: 618)
To begin you just run init(l,b) where l is the multiplier and b is the base; where I've opted for the exponential to be \(\exp(bz)\) as opposed to \(b^z\). So it's the log of our usual idea of a base. After this, mult = l and base = b and all the code will run with these values.
I've set series precision to 300 by default--this will also tell init to initialize beta with 300 iterations; you can tell init to do more iterations by writing init(l,b,count) where count is as large as you want.
There's the beta(z) which is the beta function as we know it, and there's beta_N(z) = beta(z-1/mult) which sort of guesses about where beta(0) = 1; which can be helpful.
Then I have tau which is basically as before, I have tau_N which does the same thing but with beta_N (effectively we get beta(z-1/mult) + tau(z-1/mult) = beta_N(z) + tau_N(z)).
Then I have Sexp_N(z) = beta(z) + tau(z)
Then I have some normalization protocols which I have not made perfectly. So use these are your own risk. I suggest avoiding them because I couldn't get them to work efficiently.
Then I have an error catching Sexp_Chop; which runs Sexp_N but assigns overflows to 0.
And lastly we have Mike3's graphing program.
It's still not working perfectly; but you can make some interesting graphs~!
Remember, if something isn't accurate enough; especially for exp(b) in the Shell-Thron region, you should increase the iteration depth to 1000 rather than the 300/100 I have it defaulted at. All functions which require a recursive process can always be increased adding an argument at the end telling us to go deeper. It'll slow down, but still works.
For example for base = 1/2 log(2) and mult = 1 (which is \(\sqrt{2}\) with period \(2 \pi i\)); we should increase the iteration size, displayed by this series of commands.
I'm currently making a big graph to see what this beast looks like....
I was just double checking everything before I uploaded it. It's about as good as I could get it. Here it is!
beta.gp (Size: 12.77 KB / Downloads: 618)
To begin you just run init(l,b) where l is the multiplier and b is the base; where I've opted for the exponential to be \(\exp(bz)\) as opposed to \(b^z\). So it's the log of our usual idea of a base. After this, mult = l and base = b and all the code will run with these values.
I've set series precision to 300 by default--this will also tell init to initialize beta with 300 iterations; you can tell init to do more iterations by writing init(l,b,count) where count is as large as you want.
There's the beta(z) which is the beta function as we know it, and there's beta_N(z) = beta(z-1/mult) which sort of guesses about where beta(0) = 1; which can be helpful.
Then I have tau which is basically as before, I have tau_N which does the same thing but with beta_N (effectively we get beta(z-1/mult) + tau(z-1/mult) = beta_N(z) + tau_N(z)).
Then I have Sexp_N(z) = beta(z) + tau(z)
Then I have some normalization protocols which I have not made perfectly. So use these are your own risk. I suggest avoiding them because I couldn't get them to work efficiently.
Then I have an error catching Sexp_Chop; which runs Sexp_N but assigns overflows to 0.
And lastly we have Mike3's graphing program.
It's still not working perfectly; but you can make some interesting graphs~!
Remember, if something isn't accurate enough; especially for exp(b) in the Shell-Thron region, you should increase the iteration depth to 1000 rather than the 300/100 I have it defaulted at. All functions which require a recursive process can always be increased adding an argument at the end telling us to go deeper. It'll slow down, but still works.
For example for base = 1/2 log(2) and mult = 1 (which is \(\sqrt{2}\) with period \(2 \pi i\)); we should increase the iteration size, displayed by this series of commands.
Code:
\r beta.gp
init(1,1/2*log(2))
%18 = 1.564261222210131790778549754460129428491876323129948191002835819035613273620634642498895851710066100903308133764738206355851478602526882185265925756641387118119094998243398830038599470896730999483543814425736970911144963070281914026857086826620357221541090113816887067281078711614531339222156998081447309790244319545423747866146567264533076447322117620483915175231174366379487441616911454122848179199112122759500174168803573509295856827663681833419381828025553381256620789325553560541892825360032209299559658330609591389027881708000951260708812941422980385032861100916859573318961083581807428528182941976294925218221931864653748148004212542254033819429530318333995961761644719979331349429828636607511364571319261147596784340347107762265797198610630616500412557659160530271569572410889273239386768755588383041477288488811874731173091849489782976321403004177787752092145657932236775285773138327986291591592609383932128157791680437486548324607143686079099728339143938961640299350007536653878755905191383 E-131*x^301 - 4.25210285529693790527285065730685566423856927884338819904671846018842502361777696312451939992828810846562035284285926394784958660130683312053366473488947159390779848379882423303752137192022859516295323642765662525146035309101258638630770413909736146293981604200724794713896084891832054480051256318656741860459128181083783798986278824158861233454902515606093835122406648120763847088632685421088193923678498772915198635279728440778432699242076793700148037492404851592545236134114062366606067763292167420239564631097763785673942160258571537034559977589295508953464012090592239323442023391849716699984449086891994099430629712973104765071472398242891910196951719119063709188905719229300048443937538808519146854385962753001297421400734961019460525662304690087212722407012979278965146856275695478314776002731652345905568813916198114309722509268786956901596197449718222114147676306116143692714485832928245746422734489712068280247352741023716908721658129638981348817814145897669767[+++]
/*initialize a taylor series with series precision 300 and digit precision 1000--and iteration depth the level of series precision; 300*/
exp(base*Sexp_N(20)) - Sexp_N(21)
%23 = 6.737597502028215895398620843696236117166364306438656730914692116631654872544255366451462578322003495976805126837890712397919459344514774919917508681165472836749210533903165948317745570373902391482684501540809383208322394059430181521248811175106872811147313464748579923692322028975089878286946631661798352729250155962380480386588801345611903110016812773184708048467195829968925227071197321226856514192835405255737908428990186663447493984701568953727021229903841439184400762305040795591897890343823077710282717643654799766824289427284892197695223945749475307358085395568298362956491434997806409729364223915756790083078300989399923024250388273834525408237713835620759412604059099970462993780019648963671928534089764873244635876267951224452935124979344793492129637473690923720779127195890174728413782042970011778575650217541396581866318160653584491379862020798094032665500075919122443813503166413270802019551162748562763296218553632123685926882341890037349937874415040577000264990059647 E-37
/*only 37 digit accuracy*/
exp(base*Sexp_N(20,1000)) - Sexp_N(21,1000)
%24 = 1.66171450001862563106404933114559505976485524171097378938090272158872007655879489367361586291957952497122358400290479795017486957496561408052819418173382190947343996592382708256286967388352330654947498944666671624211474620041854493853593613507116453663434430882649249952404580301675735241411577059169080541753011640861540888215334710980484257796870386621407891157469581822495578011230457960009887416702323209239718682420278604984976190997046976889281026599531814190413891776554964778698536479272100621498248789245725061768018728557071393481202535221587679833897909792732553415221205379580734473388977131011282079206469925798527095296896517035896563512705029689274313122285362234704248058642947281054062453790199008676064131931708352 E-284
/*284 digit accuracy by increasing the iteration depth*/