Fungrim entry: b63dce

Symbol: Sin —

\sin\!\left(z\right)

— Sine

The sine function

\sin\!\left(z\right)

(denoted by Sin(z) in the Fungrim formula language) is a function of a single variable. It can be defined for real and complex arguments by the series f340cb or by the differential equation 21f156 with appropriate initial values. The following table lists conditions such that Sin(z) is defined in Fungrim.

Domain	Codomain
Numbers
$z \in \mathbb{R}$	$\sin\!\left(z\right) \in \left[-1, 1\right]$
$z \in \mathbb{C}$	$\sin\!\left(z\right) \in \mathbb{C}$
Formal power series
$z \in \mathbb{R}[[x]]$	$\sin\!\left(z\right) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]]$	$\sin\!\left(z\right) \in \mathbb{C}[[x]]$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin\!\left(z\right)$	Sine
`RR`	$\mathbb{R}$	Real numbers
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`CC`	$\mathbb{C}$	Complex numbers
`FormalPowerSeries`	$K[[x]]$	Formal power series

Source code for this entry:

Entry(ID("b63dce"),
    SymbolDefinition(Sin, Sin(z), "Sine"),
    Description("The sine function", Sin(z), "(denoted by", SourceForm(Sin(z)), "in the Fungrim formula language)", "is a function of a single variable.", "It can be defined for real and complex arguments by the series", EntryReference("f340cb"), "or by the differential equation", EntryReference("21f156"), "with appropriate initial values.", "The following table lists conditions such that", SourceForm(Sin(z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, RR), Element(Sin(z), ClosedInterval(-1, 1))), Tuple(Element(z, CC), Element(Sin(z), CC)), TableSection("Formal power series"), Tuple(Element(z, FormalPowerSeries(RR, x)), Element(Sin(z), FormalPowerSeries(RR, x))), Tuple(Element(z, FormalPowerSeries(CC, x)), Element(Sin(z), FormalPowerSeries(CC, x))))))

Topics using this entry

Sine