// imports #import "@preview/drafting:0.2.2": * #import "@preview/thmbox:0.2.0": * #import "@preview/marge:0.1.0": sidenote #import "@preview/hydra:0.6.1": hydra #import "@preview/cetz:0.3.4" // math inline bounding box definitions #show math.equation.where(block: false): it => box( it, // keep the original math height: auto, // debug: //fill: red, //stroke: blue ) //#show math.equation.where(block: false): math.equation.with(block: true) // thmbox initialisation #show: thmbox-init(counter-level: 2) #let def-counter = counter("def") #show: sectioned-counter(def-counter, level: 3) #let defbox = note.with( numbering: "1.1.1", counter: def-counter, fill: rgb("#f8f8f8"), border: (paint: rgb("#787878"), thickness: 0.8pt), radius: 3pt, inset: (x: 0.8em, y: 0.6em), ) // link colours #show link: set text(fill: blue) #show link: underline // convenience math commands / aliases #let iff = $<==>$ #let imp = $==>$ #let ve(body) = { $op(upright(bold(body)))$ } #let abs(body) = $bar.v #body bar.v$ #let norm(body, p: none) = { if p == none { $bar.v.double #body bar.v.double$ } else { $bar.v.double #body bar.v.double_(#p)$ } } #let scr(it) = text( features: ("ss01",), box($cal(it)$), ) // font sizes #show math.equation: set text(14pt) #set text( size: 14pt ) #set document(title: "Catalogue of Scalar Valued Functions", author: "Aayush Bajaj") #align(center)[ #block(text(weight: "bold", size: 24pt)[Scalar Valued Functions]) #v(1em) #text(weight: "bold", size: 20pt)[Aayush Bajaj] #v(1em) #text(size: 16pt)[Version 0.1] #v(1em) #text(size: 16pt)[#datetime.today().display()] #v(1em) #set par(justify: false) *Abstract* \ Whilst the author believes the raison d'ĂȘtre of this manuscript is obvious, they do not believe that the scope is. The _Theory of Functions_ is rich and central to Mathematics. As such, we limit our scope here to definitions and graphs of univariate functions $f: RR -> RR$. Whilst we include common equalities between different functions - say circular and exponential - what you will not find here are derivations of any sort. You will *not* find proofs *nor* set theoretic discussions of "jectivities", binary relations, etc. Furthermore there is a purposeful lack of rigour in this /catalogue/; theorems are asserted as is, with no warranty and no proof. Finally, analytic concerns of limits and convergence are also dutifuly ignored. #image("crest.svg", width: 60%) ] #pagebreak() #set page( numbering: "1", ) #outline(title: "Table of Contents") #pagebreak() #set heading(numbering:"1.") #set math.equation(numbering: "(1)") #set page( header: context{ align(left, emph(hydra(1))) } ) // some reference ways: @algebraic[words] ; #ref(); #link()[word] = Elementary These such functions are continuous on their domains and include taking *sums*, *products*, *roots* and *compositions* of finitely many #link()[algebraic] or #link()[transcendental] functions. == Algebraic === Polynomials $ p(x) &= a_n x^n + a_(n-1)x^(n-1) + dots.h + a_2 x^2 + a_0 &= sum^n_(k=0) a_k x^k $ // image === Rational Much in the same way that $QQ$ is defined as any element $a/b$ where $a,b in ZZ$: #align(center)[#image("sets.svg", width: 30%)] a function $f$ is called a rational function if it can be written in the form: $ f(x) = (P(x))/(Q(x)) $ where $P(x)$ and $Q(x)$ are polynomial functions of $x$ and $Q$ is not the zero function. // image === Power Note that $sqrt(x)$ is not a polynomial because $sqrt(x) = x^(1/2)$ and $1/2 in.not ZZ$. // image == Transcendental These are the analytic functions that *do not* satisfy a polynomial equation. === Exponential $ e = lim_(n -> infinity) (1 + 1/n)^n $ furthermore, $ exp(x) = lim_(n -> infinity) (1 + x/n)^n $ graphically we have: // image and by Euler's identity we have: $ e^(i theta) = cos(theta) + i sin(theta) $ which relates the "circular" functions cosine and sine with the "exponential" $square$ === Logarithm setting $y = e^x$ and swapping variables: $x = e^y imp y = ln(x)$. as such the logarithm and exponential functions are inverses of each other. // image inverse === Trigonometric === Inverse Trig === Reciprocal Trig === Hyperbolic === Inverse Hyper === Reciprocal Hyper === Factorial $x!$ and $1/(x!)$ #pagebreak() = Non-Elementary == Transcendental === Gamma === Beta === Riemann Zeta === Error $ op("erf")(x) = 2 / sqrt(Pi) integral^x_0 e^((-t)^2) dif t $ === Tetration === Elliptic Integrals === Trigonometric Integrals $ op("Si")(x) = integral_0^x (sin(t))/t dif t $ $ op("si")(x) = - integral^x_infinity (sin(t))/t dif t $ $ op("Si") - op("si") = Pi/2 = integral^infinity_0 (sin(t))/t dif t $ label as Dirichlet's integral === Fresnel $ op("S")(x) = integral^x_0 sin(t^2) dif t, thick op("C")(x) = integral^x_0 cos(t^2) dif t $ == Algebraic === Bessel === Hypergeometric $ Beta_0 + Beta_1 z + Beta_2 z^2 + dots.h = sum_(n>=0) Beta_n z^n $ where the ratio of successive coefficients is a rational function of $n$: $ (Beta_(n+1))/(Beta_n) = (A(n))/(B(n)) $ #pagebreak() = Discontinuous == Absolute Value == Step === Heaviside === Floor === Ceiling === Square Wave