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The Mathematical Functions Grimoire

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Welcome! The Mathematical Functions Grimoire (Fungrim) is an open
source library of formulas and data for special functions. Fungrim
currently consists of 457 symbols (named mathematical objects), 3130
entries (definitions, formulas, tables, plots), and 82 topics
(listings of entries). This is one example entry:

9ee8bc
Details
z [?](s)=2(2p)s-1sin[?] [?](ps2)G [?](1-s)z [?](1-s)\zeta\!\left(s\right) = 2
{\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \
Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)z(s)=2(2p)s-1sin(2
ps )G(1-s)z(1-s)
Assumptions:s[?]C  and[?]  s[?]Z>=0s \in \mathbb{C} \;\mathbin{\operatorname
{and}}\; s \notin \mathbb{Z}_{\ge 0}s[?]Cands[?]/ Z>=0 
Alternative assumptions:s[?]C[[x]]  and[?]  s[?]Z>=0s \in \mathbb{C}[[x]] \;
\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}s[?]C[[x]]and
s[?]/ Z>=0 
TeX:

\zeta\!\left(s\right) = 2 {\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}

s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \notin \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol         Notation               Short description
RiemannZeta    z [?](s)\zeta\!\left(s\     Riemann zeta function
               right)z(s)
Pow            ab{a}^{b}ab               Power
Pi             p\pip                     The constant pi (3.14...)
Sin            sin[?](z)\sin(z)sin(z)      Sine
Gamma          G(z)\Gamma(z)G(z)         Gamma function
CC             C\mathbb{C}C              Complex numbers
ZZGreaterEqual Z>=n\mathbb{Z}_{\ge n}Z>=n  Integers greater than or
                                         equal to n
PowerSeries    K[[x]]K[[x]]K[[x]]        Formal power series

Source code for this entry:

Entry(ID("9ee8bc"),
    Formula(Equal(RiemannZeta(s), Mul(Mul(Mul(Mul(2, Pow(Mul(2, Pi), Sub(s, 1))), Sin(Div(Mul(Pi, s), 2))), Gamma(Sub(1, s))), RiemannZeta(Sub(1, s))))),
    Variables(s),
    Assumptions(And(Element(s, CC), NotElement(s, ZZGreaterEqual(0))), And(Element(s, PowerSeries(CC, SerX)), NotElement(s, ZZGreaterEqual(0)))))

The Fungrim website provides a permanent ID and URL for each entry,
symbol or topic. Click "Details" to show an expanded view of an
entry, or click the ID (9ee8bc) to show the expanded view on its own
page. All data in Fungrim is represented in semantic form designed to
be usable by computer algebra software.

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided
under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC