update documentation

Author: danielsuit

README.md

@@ -7,54 +7,136 @@ Run ```pip install mathfunctionize``` to get started
 
 Import the library into python file: ```from mathfunctionize import mathfunctionize```
 
-Functions are able to be used by stating ```mathfunctionize.intendedFunction()``` and replacing ```intendedFunction()``` with the one of the avalible functions listed below.
+Functions are able to be used by stating ```mathfunctionize.intendedFunction()``` and replacing ```intendedFunction()``` with one of the available functions listed below.
 
-## Operations:
+## Constants
 ---
-```addition(a, b)``` - takes two numerical inputs and returns the sum of the two numbers
+```pi``` - the ratio of a circle's circumference to its diameter, approximately 3.14159
 
-```subtraction(a, b)``` - takes two numerical inputs and returns the number minus the second number
+```e``` - Euler's number, the base of the natural logarithm, approximately 2.71828
 
-```multiplication(a, b)``` - takes two numerical inputs and returns the product of the two numbers
+## Arithmetic
+---
+```addition(arr)``` - takes a list of numbers and returns the sum of all elements, e.g. ```addition([1, 2, 3])``` returns ```6```
+
+```subtraction(arr)``` - takes a list of numbers and subtracts each subsequent element from the first, e.g. ```subtraction([10, 3, 2])``` returns ```5```
+
+```multiplication(arr)``` - takes a list of numbers and returns the product of all elements, e.g. ```multiplication([2, 3, 4])``` returns ```24```
+
+```division(arr)``` - takes a list of numbers and divides the first element by each subsequent element, e.g. ```division([100, 5, 2])``` returns ```10```
+
+```power(arr)``` - takes a list of numbers and exponentiates left-to-right, e.g. ```power([2, 3])``` returns ```8```
+
+```modulo(arr)``` - takes a list of numbers and applies the modulo operator left-to-right, e.g. ```modulo([10, 3])``` returns ```1```
+
+```flatDivision(arr)``` - takes a list of numbers and performs floor division left-to-right, e.g. ```flatDivision([10, 3])``` returns ```3```
+
+## Algebra
+---
+```gamma(x)``` - computes the gamma function of x, a generalization of the factorial to real and complex numbers where gamma(n) = (n-1)! for positive integers
+
+```factorial(x)``` - returns the factorial of a non-negative integer x, i.e. x! = x * (x-1) * ... * 1
+
+```absolute(x)``` - returns the absolute value of x
+
+```squareRoot(x)``` - returns the square root of x
+
+```cubeRoot(x)``` - returns the cube root of x
+
+```nthRoot(x, n)``` - returns the nth root of x
+
+```round(x, place)``` - rounds x to the nearest specified place value (e.g. 1, 10, 100)
+
+## Counting
+---
+```combinations(n, r)``` - returns the number of ways to choose r items from n items without regard to order, i.e. n! / (r! * (n-r)!)
+
+```permutations(n, r)``` - returns the number of ways to arrange r items from n items where order matters, i.e. n! / (n-r)!
+
+```circularPermutations(n)``` - returns the number of ways to arrange n items in a circle, i.e. (n-1)!
+
+```derangements(n)``` - returns the number of permutations of n elements where no element appears in its original position
+
+## Probability
+---
+```bayes_theorem(priorA, priorB, likelihoodA, likelihoodB)``` - computes the posterior probability using Bayes' theorem given prior probabilities and likelihoods for two hypotheses
 
-```division(a, b)``` - takes two numerical inputs and returns the first number divided by the second number
+```uniformPDF(a, b)``` - returns the probability density for a continuous uniform distribution over the interval [a, b]
 
-```power(a, b)```
+```uniformCDF(X, a, b)``` - returns the cumulative distribution function value for a uniform distribution, giving the probability that a random variable is less than or equal to X
 
-```modulo(a, b)```
+```normalPDF(X, mean, stdDev)``` - returns the probability density of the normal (Gaussian) distribution at point X given a mean and standard deviation
 
-```flatDivision(a, b)```
+```normalCDF(x, mean, stdDev)``` - returns the cumulative distribution function value for the normal distribution, approximating the probability that a random variable is less than or equal to x
 
-```factorial(x)```
+```gammaPDF(x, a, b)``` - returns the probability density of the gamma distribution at point x with shape parameter a and rate parameter b
 
-## Mathematics
+## Complex Numbers
 ---
-```absolute(x)```
+```complex_addition(a, b)``` - adds two complex numbers represented as strings (e.g. ```"3+4i"```, ```"2-1i"```) and returns the result as a string
 
-```round(x, place)```
+```complex_subtraction(a, b)``` - subtracts two complex numbers represented as strings and returns the result as a string
 
-## Quantative Analysis
+## Trigonometry
 ---
-```localMinimum(arr)```
+```sin(x)``` / ```sine(x)``` - returns the sine of x (in radians) using a Taylor series approximation
 
-```localMaximum(arr)```
+```cos(x)``` / ```cosine(x)``` - returns the cosine of x (in radians) using a Taylor series approximation
 
-```globalMinimum(arr)```
+```tan(x)``` / ```tangent(x)``` - returns the tangent of x (in radians), computed as sine(x) / cosine(x)
 
-```globalMaximum(arr)```
+```csc(x)``` / ```cosecant(x)``` - returns the cosecant of x (in radians), computed as 1 / sine(x)
+
+```sec(x)``` / ```secant(x)``` - returns the secant of x (in radians), computed as 1 / cosine(x)
+
+```cot(x)``` / ```cotangent(x)``` - returns the cotangent of x (in radians), computed as cosine(x) / sine(x)
+
+```arcsine(x)``` - returns the inverse sine of x (where -1 <= x <= 1) using a Taylor series approximation
+
+```arccosine(x)``` - returns the inverse cosine of x (where -1 <= x <= 1) using a Taylor series approximation
+
+```arctangent(x)``` - returns the inverse tangent of x (where -1 <= x <= 1) using a Taylor series approximation
+
+```arccotangent(x)``` - returns the inverse cotangent of x (x cannot be 0)
+
+```arcsecant(x)``` - returns the inverse secant of x (where |x| >= 1)
+
+```arccosecant(x)``` - returns the inverse cosecant of x (where |x| >= 1)
+
+```degreeToRadian(degree)``` - converts an angle from degrees to radians, normalizing to the range [0, 2*pi]
+
+```radianToDegree(radian)``` - converts an angle from radians (in the range [0, 2*pi]) to degrees
+
+## Quantitative Analysis
+---
+```localMinimum(arr)``` - finds all local minima in an array and returns a list containing the count and their positions
+
+```localMaximum(arr)``` - finds all local maxima in an array and returns a list containing the count and their positions
+
+```globalMinimum(arr)``` - finds the smallest value in an array and returns a list containing the value and all positions where it occurs
+
+```globalMaximum(arr)``` - finds the largest value in an array and returns a list containing the value and all positions where it occurs
 
 ## Statistics
 ---
-```mean(arr)```
-```median(arr)```
-```standardDevation(arr)```
-```mode(arr)```
+```mean(arr)``` - returns the arithmetic mean (average) of a list of numbers
+
+```median(arr)``` - returns the median (middle value) of a sorted list of numbers, averaging the two middle values for even-length lists
+
+```standardDevation(arr)``` - returns the population standard deviation of a list of numbers, measuring the spread of data around the mean
+
+```mode(arr)``` - returns the most frequently occurring value in a list of numbers
+
+```variance(arr)``` - returns the population variance of a list of numbers, measuring how far each value is from the mean
 
 ## Linear Algebra
-```additionMatrix(arr)```
-```subtractionMatrix(arr)```
-```multiplicationMatrix(arr)```
-```determinant(arr)```
-```transpose(arr)```
+---
+```additionMatrix(arr1, arr2)``` - adds two matrices of the same dimensions element-wise and returns the resulting matrix
+
+```subtractionMatrix(arr1, arr2)``` - subtracts two matrices of the same dimensions element-wise and returns the resulting matrix
+
+```multiplicationMatrix(arr1, arr2)``` - performs matrix multiplication on two matrices where the number of columns in arr1 equals the number of rows in arr2
 
+```determinant(arr)``` - computes the determinant of a square matrix using cofactor expansion
 
+```transpose(arr)``` - returns the transpose of a matrix, swapping rows and columns

upcoming.md

@@ -1,94 +1,60 @@
 # List of upcoming functions to be added
 ---
 
-## Trigonometry
-
-sin()
-
-cos()
-
-tan()
-
-arcsin()
-
-arccos()
-
-arctan()
-
-sec()
-
-csc()
-
-cot()
-
 ## Metric Spaces
 
-isMetricSpace()
+`isMetricSpace(d, S)` - Determines whether a given distance function d satisfies the three metric space axioms (non-negativity with identity of indiscernibles, symmetry, and the triangle inequality) over a set S. Returns a boolean indicating whether (S, d) forms a valid metric space.
 
-dist()
+`dist(x, y, metric)` - Computes the distance between two points x and y under a specified metric. Supports common metrics such as Euclidean, Manhattan, and Chebyshev distance in n-dimensional space.
 
 ## Calculus
-limit()
 
-derivitive()
+`limit(f, x, a)` - Evaluates the limit of a function f(x) as x approaches a value a, using numerical approximation from both the left and right sides. Detects one-sided limits and indicates when a two-sided limit does not exist.
+
+`derivative(f, x)` - Computes the numerical derivative of a function f at a given point x using finite difference approximation. Returns the instantaneous rate of change of f at x.
 
-concavity()
+`concavity(f, x)` - Determines the concavity of a function f at a point x by computing the second derivative. Returns whether the function is concave up (positive second derivative), concave down (negative second derivative), or at an inflection point.
 
-integral()
+`integral(f, a, b)` - Computes the definite integral of a function f over the interval [a, b] using numerical integration (e.g. Simpson's rule or the trapezoidal rule). Returns the signed area under the curve.
 
-continuity()
+`continuity(f, x)` - Tests whether a function f is continuous at a point x by checking that the left-hand limit, right-hand limit, and function value at x all exist and are equal. Returns a boolean.
 
 ## Complex Analysis
 
-conjugate()
+`conjugate(z)` - Returns the complex conjugate of a complex number z. For z = a + bi, the conjugate is a - bi. Takes a string representation and returns a string.
 
-rootsOfUnity()
+`rootsOfUnity(n)` - Computes all n-th roots of unity, i.e. the n complex numbers z such that z^n = 1. Returns them as a list of complex number strings evenly spaced on the unit circle.
 
 ## Set Theory
 
-powerSet()
+`powerSet(S)` - Returns the power set of a given set S, which is the set of all possible subsets of S including the empty set and S itself. For a set of size n, the power set contains 2^n elements.
 
-union()
+`union(A, B)` - Returns the union of two sets A and B, containing all elements that are in A, in B, or in both.
 
-intersection()
+`intersection(A, B)` - Returns the intersection of two sets A and B, containing only the elements that are present in both A and B.
 
-isOpenSet()
+`isOpenSet(S, topology)` - Determines whether a given subset S is an open set within a specified topology. Returns a boolean based on whether S belongs to the collection of open sets defined by the topology.
 
 ## Real Analysis
 
-## Statistics
-
-nPr()
-
-nCr()
-
 ## Number Theory
 
-isPrime(x)
+`isPrime(x)` - Determines whether a positive integer x is a prime number, i.e. a number greater than 1 whose only divisors are 1 and itself. Returns a boolean.
 
 ## Topology
 
-smooth()
+`smooth(f, x)` - Tests whether a function f is infinitely differentiable (smooth) at a point x by computing successive numerical derivatives and checking for convergence. Returns a boolean or a smoothness classification.
 
 ## Polynomials
-factor()
 
-divide()
+`factor(coefficients)` - Factors a polynomial given its coefficients into irreducible polynomial factors. Returns the factored form as a list of factor-multiplicity pairs.
 
-zeros()
-## Linear Algebra
+`divide(dividend, divisor)` - Performs polynomial long division on two polynomials represented by their coefficient lists. Returns the quotient and remainder as separate coefficient lists.
 
-powerSet()
-
-union()
-
-intersection()
-
-isOpenSet()
+`zeros(coefficients)` - Finds all real and complex roots (zeros) of a polynomial given its coefficients. Returns a list of values where the polynomial evaluates to zero.
 
 ## Knot Theory
 
 ## Type Theory
 
 ## Homotopy Theory
-