# Fungrim entry: b63dce

Symbol: Sin $\sin(z)$ Sine
The sine function $\sin(z)$ (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
$z \in \mathbb{R}$ $\sin(z) \in \left[-1, 1\right]$
$z \in \mathbb{C}$ $\sin(z) \in \mathbb{C}$
Formal power series
$z \in \mathbb{R}[[x]]$ $\sin(z) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]]$ $\sin(z) \in \mathbb{C}[[x]]$
Table data: $\left(P, Q\right)$ such that $\left(P\right) \;\implies\; \left(Q\right)$
Definitions:
Fungrim symbol Notation Short description
Sin$\sin(z)$ Sine
RR$\mathbb{R}$ Real numbers
ClosedInterval$\left[a, b\right]$ Closed interval
CC$\mathbb{C}$ Complex numbers
PowerSeries$K[[x]]$ 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, PowerSeries(RR, x)), Element(Sin(z), PowerSeries(RR, x))), Tuple(Element(z, PowerSeries(CC, x)), Element(Sin(z), PowerSeries(CC, x))))))

## Topics using this entry

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