Functions
In the old saying, “The death of a mathematician means the loss of a function,” it encompasses functions used across the entirety of mathematics, too versatile to be confined to just one category.
- Arithmetic functions used in analytic number theory are separated into the Number Theory category.
- Activation functions frequently used in deep learning are placed in the Machine Learning category.
- Boolean functions are placed in the Quantum Information Theory category.
Anonymous Functions
- Constant function $c$
- Polynomial function $P(z)$
- Even and odd functions
- Generating function
- Periodic function
- Harmonic function $\Delta f = 0$
- Linear function
- Additive and multiplicative functions
- Composite function $f \circ g$
- Monotonic, increasing, and decreasing functions
- General convex functions
- Function restriction and extension $f\vert_{U}, \tilde{f}$
- Inclusion function $i$
- Alternating function
- Matrix functions, exponential matrix functions $\mathbf{x}(t) , A(t)$
- Radial function
- Bounded function
- Multivalued mapping $f : X \rightrightarrows Y$
- Homogeneous function $f(ax) = a^{n}f(x)$
- Variable Separable Function $f(x, y) = g(x)h(y)$
Named Functions
- Ceiling and floor functions $\lceil \cdot \rceil$, $\lfloor \cdot \rfloor$
- Diagonal matrix as a function, diagonal elements $\text{diag}$
- Exponential function $\exp$
- Logarithm function $\log$
- Absolute value function $\left| \cdot \right|$
- Step function $H$
- Hard thresholding and soft thresholding $\eta$
Trigonometric Functions
- Trigonometric functions $\sin$, $\tan$, $\sec$
- Exact trigonometric values at common angles
- Addition theorems
- Proof of $\sin^2 + \cos^2 = 1$
- Law of cosines
- Sum-to-product and product-to-sum formulas
- Half-angle and double-angle formulas
- Composition formula $A\cos\theta + B \sin \theta = \sqrt{A^{2} + B^{2}}\sin(\theta + \phi)$
- Identities
- Derivatives
- Translation and Derivative Relationships of Trigonometric Functions
- Various Integration Methods for Trigonometric Functions
- Inverse trigonometric functions $\sin^{-1}$, $\tan^{-1}$
- Hyperbolic functions $\sinh$, $\tanh$
- Inverse hyperbolic functions $\sinh^{-1}, \cosh^{-1}, \tanh^{-1}$
- Sinc function $\operatorname{sinc}$
Alphabet Functions
Gamma Function
- Factorial, double factorial, multifactorial
- Gamma function $\Gamma$
- Euler’s limit formula: Second form of the gamma function
- Weierstrass’s infinite product: Third form of the gamma function
- Euler’s reflection formula
- Legendre’s duplication formula
- Rigorous proof of Stirling’s approximation formula
- Digamma function: Derivative of the gamma function and its reciprocal $\psi_{0} := \Gamma’ / \Gamma$
Beta Function
Riemann Zeta Function
- Riemann zeta function $\zeta$
- Dirichlet eta function $\eta$
- Relationship between the gamma function, Riemann zeta function, and Dirichlet eta function
- Poisson summation formula
- Jacobi theta function $\vartheta$
- Riemann Xi function $\xi$
- Riemann functional equation and trivial zeros of the Riemann zeta function
- Riemann hypothesis
- Ramanujan summation
Special Functions
Bessel Functions
- Bessel function $I_{\nu}$
- Neumann function, Bessel function of the second kind $N_{\nu}$, $Y_{\nu}$
- Hankel functions, Bessel function of the third kind $H_{\nu}$
- Modified Bessel equation and modified Bessel functions $I_{\nu}$, $K_{\nu}$
Legendre Polynomials
Hermite Polynomials
Miscellaneous
- Why ‘implicit function’ is a mistranslation
- The history of the Dirac delta function and why Dirac used the delta function
All posts
- Special Angles of Trigonometric Functions
- Notation and Naming Conventions of Hyperbolic Functions
- The Relationship between the Translation of Trigonometric Functions and Their Derivatives
- Integration Techniques for Various Trigonometric Functions
- Odd Functions and Even Functions
- Addition Formula for Trigonometric Functions: Various Proofs
- Gamma Function
- Dirac Delta Function
- Properties of the Dirac Delta Function
- Euler's Limit Formula Derivation for the Gamma Function
- Proof of the Convergence of the Euler-Mascheroni Constant
- Weierstrass's Infinite Product for the Gamma Function
- Euler's Reflection Formula Derivation
- Proof of Euler's Representation of the Sinc Function
- What is a Generating Function?
- Convex Functions, Concave Functions
- Euler's Proof: Finding the Sum of Reciprocals of Squares Using the Sinc Function
- Generalization of Binomial Coefficients: Beta Function
- Trigonometric Representation of the Beta Function
- Double Angle and Half Angle Formulas of Trigonometric Functions
- Proof that Sine Squared Plus Cosine Squared Equals 1
- Derivation of the Legendre Duplication Formula
- Sum and Difference Formulas and Product-to-Sum Formulas of Trigonometric Functions
- Why is "Implicit Function" a Misleading Translation?
- Wallis Product
- Rigorous Proof of Stirling's Approximation Formula
- Staircase Function
- Rodrigues Formula for Legendre Polynomial
- Legendre Polynomials are orthogonal to any lower degree polynomial
- Orthogonality of Legendre Polynomials
- Proof of the Second Cosine Law Using the Definition of Trigonometric Functions
- The Integral of a Periodic Function Over One Period is Constant Regardless of the Integration Interval
- Derivation of the Base Change Formula for Logarithms
- Any Function Can Always Be Expressed as the Sum of Odd and Even Functions
- Half-Wave Symmetric Function
- Additive and Multiplicative Functions
- Properties of Continuous Functions with Additivity
- Semi-linear Function
- Matrix Functions, Definition of Matrix Exponential Functions
- Derivation of the Gamma Function
- Factorial, Double Factorial, and Multifactorial
- Various Important Formulas Involving the Gamma Function and Factorials
- Relationship between Beta Function and Gamma Function
- Representation of the Beta Function in the Form of an Improper Integral
- Euler Integrals: Beta Function and Gamma Function
- Recursive Relations of Legendre Polynomials
- Generating Functions of Legendre Polynomials
- Legendre Polynomials
- Associated Legendre Polynomials for Negative Index m
- Orthogonality of Associated Legendre Polynomials
- Associated Legendre Polynomials
- Bessel Function's Recursive Relations
- Orthogonality of Bessel Functions
- Bessel Functions
- Hankel Functions, Bessel Functions of the Third Kind
- Modified Bessel Equation and Modified Bessel Function
- The Airy Function
- Hermite Functions
- Foehammer Symbol
- Hermite Polynomials
- Hermite Polynomials' Generating Function
- Hermite Polynomials' Recursive Relations
- Orthogonality of Hermite Polynomials
- Laguerre Polynomials' Rodrigues' Formula
- Riemann Zeta Function
- Dirichlet eta function
- Relationship between the Gamma Function and the Riemann Zeta Function and the Dirichlet Eta Function
- Hyperbolic Functions' Identities
- Proof of the Addition Formulas for Hyperbolic Functions
- Double and Half Angle Formulas for Hyperbolic Functions
- Hyperbolic Functions
- Sum and Difference Formulas and Multiplication Formulas for Hyperbolic Functions
- Jacobi Theta Function
- Riemann Zeta Function
- The History of the Delta Function and Why Dirac Used the Delta Function
- Riemann Hypothesis and Trivial Roots of the Riemann Zeta Function
- Characteristic Function, Indicator Function
- Monotonic Functions, Increasing Functions, Decreasing Functions
- Semi-Linear (Conjugate Linear) Functions
- Riemann Hypothesis
- Derivation of the Laurent Expansion of the Riemann Zeta Function
- Ramanujan Sum
- Analytic Proof that 1+1+1+1+1+⋯=-1/12
- Linear Function
- Analytic Proof of 1+2+3+4+5+⋯=-1/12
- Derivatives of Logarithmic Functions
- Composition of Functions
- Differentiation of Exponential Functions
- Limits of Exponential and Logarithmic Functions
- Ceiling Function and Floor Function
- Diagonal Matrix as a Function, Diagonal Elements
- Periodic Function
- Absolute Value Function
- Survival Function
- Definition of Trigonometric Functions
- Polynomial Function
- Exponential Functions
- Definition of Logarithmic Functions
- Formulas Related to Factorials
- Harmonic Function
- Inverse Trigonometric Functions
- Expansion and Contraction of a Function
- Including Functions
- Alternating Function
- Sign function
- Identity Function
- Ramp Function
- Various Properties of Convex Functions
- Sum and Difference Identities for Trigonometric Functions
- Hyperbolic Functions Composite Formula
- Definition of a General Convex Function
- Definition of a Rational Function
- Radial Functions
- Definition of a Constant Function
- Bounded Function
- Homogeneous Function
- Differentiation of Trigonometric Functions
- Why Factorial 0 is Defined as 0!=1
- Biharmonic Functions
- Polyharmonic Functions
- Definition of the Arctan2 Function
- Hard Thresholding and Soft Thresholding as Functions
- General Polyhedral Mapping, Definition of Set-Valued Mapping
- What is a Special Function?
- Rodrigues' Formula for Multiple Polynomials
- Laguerre Polynomial
- Rodrigues' Formula for Hermite Polynomials
- Variables Separable Function
- Reciprocal times Derivative of Gamma Function
- Derivatives of Inverse Trigonometric Functions
- Trigonometric Identities
- Derivative of Gamma Function at 1
- Derivatives of Hyperbolic Functions
- Inverse Hyperbolic Functions
- Derivative of the Inverse Hyperbolic Functions
- Simple Poles of Gamma Function