analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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symmetric monoidal (∞,1)-category of spectra
See at trigonometric function.
The cosine function is the function $\cos \;\colon\; \mathbb{R} \to \mathbb{R}$ from the real numbers to themselves which is characterized by the following equivalent conditions:
$\cos$ is the unique solution among smooth functions to the differential equation/initial value problem
(where a prime indicates the derivative) subject to the initial conditions
(Euler's formula) $\cos$ is the real part of the exponential function with imaginary argument
$\cos$ is the unique function given by the infinite series
In general, Archimedean ordered fields which are not sequentially Cauchy complete do not have an cosine function. Nevertheless, the cosine map is still guaranteed to be a partial function, because every Archimedean ordered field is a Hausdorff space and thus a sequentially Hausdorff space. Therefore an axiom could be added to Archimedean ordered fields $F$ to ensure that the cosine partial function is actually a total function:
Axiom of cosine function: For all elements $x \in F$, there exists a unique element $\cos(x) \in F$ such that for all positive elements $\epsilon \in F_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!}\right) - \cos(x) \lt \epsilon$.
There is another axiom which uses the fact that derivatives of functions are well defined in the ordered local Artin $F$-algebra $F[\epsilon]/\epsilon^2$ by the equation $f(x + \epsilon) = f(x) + f'(x) \epsilon$:
Axiom of sine and cosine functions: Let $F[\epsilon]/\epsilon^2$ be the ordered local Artin $F$-algebra, with non-zero non-positive non-negative nilpotent element $\epsilon \in F[\epsilon]/\epsilon^2$ where $\epsilon^2 = 0$ and canonical $F$-algebra homomorphism $h:F \to F[x]/x^2$. There exists unique functions $\sin:F \to F$ and $\cos:F \to F$ and functions $\sin':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ and $\cos':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ such that for every element $x \in F$, $h(\sin(x)) = \sin'(h(x))$, $h(\cos(x)) = \cos'(h(x))$, $\sin'(x + \epsilon) = \sin'(x) + \cos'(x) \epsilon$, $\cos'(x + \epsilon) = \cos'(x) - \sin'(x) \epsilon$, $\sin'(0) = 0$, and $\cos'(0) = 1$.
The dual number real algebra $\mathbb{D} \coloneqq \mathbb{R}[\epsilon]/\epsilon^2$ has a notion of sine and cosine function, which are the solutions to the system of functional equations $\sin(x + \epsilon) = \sin(x) + \cos(x) \epsilon$ and $\cos(x + \epsilon) = \cos(x) - \sin(x) \epsilon$ with $\sin(0) = 1$ and $\cos(0) = 1$.
In classical mathematics, one could prove that the modulated Cantor real numbers $\mathbb{R}_C$ are sequentially Cauchy complete and equivalent to the HoTT book real numbers $\mathbb{R}_H$. However, in constructive mathematics, the above cannot be proven; while the HoTT book real numbers $\mathbb{R}_H$ are still sequentially Cauchy complete, the modulated Cantor real numbers $\mathbb{R}_C$ in general cannot be proven to be sequentially Cauchy complete. In particular, this means that the sequence
does not have a limit for all modulated Cantor real numbers $x \in \mathbb{R}_C$. However, the sequence, by definition of $\mathbb{R}_C$, does have a limit for all rational numbers $x \in \mathbb{Q}$; this means that one could restrict the domain of the cosine function to the rational numbers $\cos:\mathbb{Q} \to \mathbb{R}_C$, and define it in the usual manner:
Discussion in constructive analysis:
Errett Bishop, §7 in: Foundations of Constructive Analysis, McGraw-Hill (1967)
Errett Bishop, Douglas Bridges §7 in: Constructive Analysis, Grundlehren der mathematischen Wissenschaften 279, Springer (1985) [doi:10.1007/978-3-642-61667-9]
See also:
Last revised on February 8, 2023 at 19:43:01. See the history of this page for a list of all contributions to it.