Euler modular function note

The Euler Q function is represented by the following infinite product, also denoted by the Q-Pochhammer symbol.

Wikipedia

Numerical results on the real axis.

Variable transformation yields a bounded curve suitable for numerical evaluation.

Table(digit:300)
Limit Table(digit:75)

q = -0.
679334369989992724165789016984783579936441256412012010234918
431856742847762651599470526827588108924759171702586718086099
369463758389555802272035707544592279109456240352614582155450
616435344973909866235729991887465528174068568693290066812447
473189502504504553741300250987237061605868316567793854504189
978430035639448527006103817728648448250002809851221265884712
437739302756785855529694137879480078350051903905975035842033
083686415524254448747387606693269131835790825535265787878936
804387088060534411032905874676308304084197967426302028245044
037141868160535979777123048756733104631979868660533830616924
930801338945984648674439977350829241590167608706550761787998...

q_max = -0.
411248479177954773444025662435572436972040503633601105570211
017836442913453381447150772095063339241856731081456480033459
519730194900584872796235280441425981938799877850007894346946
199232162562462179351528160789438258962671492753248122230456
618610661212942585881017690448278685179049368390281841409428
610969710907133689148278526357780142440691979016743813743352
252789328025504742285389994624351008276745144314836961579225
497943232385778700278841196745374319313994311736634968184193
435040375104670922277523631959434304376996855157970358945657
234672551713535156163704300160124548700597632973242300954367...

OEIS A143441

Φ(q_max) = 1.
228348867038575112586878389860096824990327915769526746796298
315103082545177832886480199936324256119640778119785800277975
675603915212028749801283037896712222597382203127470933035372
330162531048008378590092516435578593402681064591620502130889
990356580002956439973063121793379466182876335977390999433307
439811703127753282960831762226681916066105169870906927419688
341239592282619416450353516300860064946572441110381338044510
018369830081014227187541445541694168504972335368709033040356
094873998958489011641170140913194320878622304051921800210335
035857471976383959541758533602002933138606004469363730348519...

OEIS A143440

The series is shown in the following equation using the divisor function σ₁(n). OEIS A000203

The infinite sum of this series at |q|=1 can be evaluated from the limit values as follows.

Abs	Arg

Abs	Arg

In |q|=1, essential singularities are arranged in a regular pattern based on Dedekind eta function.

FP64 Source