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 $P(z)$
  • Rational function $Q(z)$
• Even and odd functions
  • Any function can always be expressed as the sum of an even and an odd function
• Generating function
• Periodic function
  • The integral of a periodic function over one period is always the same, regardless of the integration interval
  • Half-wave symmetric function
• Harmonic function $\Delta f = 0$
  • Biharmonic function $\Delta^{2} f = 0$
  • Polyharmonic function $\Delta^{k} f = 0$
• Linear function
  • Quasilinear function
  • Conjugate linear function
• Additive and multiplicative functions
  • Properties of continuous functions with additivity
• Composite function $f \circ g$
• Monotonic, increasing, and decreasing functions
  • Survival function
• General convex functions
  • Convex and concave functions
  • Various properties of 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 $\exp$
  • Differentiation of the exponential function
• Logarithm function $\log$
  • Base Change Formula for Logarithms
  • Differentiation of the logarithm function
  • Limits of the exponential and logarithm functions
• Absolute value function $\left| \cdot \right|$
• Step function $H$
• Hard thresholding and soft thresholding $\eta$
  • Ramp function $R$
  • Sign function $\text{sgn}$
  • Identity function $I, \text{1}, \text{id}$

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 $\sin^{-1}$, $\tan^{-1}$
  • Arctangent2 $\arctan 2$
  • Derivatives
• Hyperbolic functions $\sinh$, $\tanh$
  • Notation and Naming Conventions
  • Addition theorems
  • Sum-to-product and product-to-sum formulas
  • Half-angle and double-angle formulas
  • Composition formula $A\cosh\theta + B \sinh\theta = \sqrt{A^{2} - B^{2}}\sinh(\theta + \phi)$
  • Identity formulas
  • Derivatives
• Inverse hyperbolic functions $\sinh^{-1}, \cosh^{-1}, \tanh^{-1}$
  • Derivatives
• Sinc function $\operatorname{sinc}$
  • Euler’s proof: Using the sinc function for the sum of the reciprocals of squares
  • Wallis product

Alphabet Functions

• Dirac delta function $\delta$
  • Properties
• Characteristic function, indicator function $\chi$

Gamma Function

• Factorial, double factorial, multifactorial
  • Formulas involving factorials
  • Why factorial of 0 is defined as 0!=1
• Gamma function $\Gamma$
  • Derivation of the gamma function
  • Pochhammer symbol
  • Various important formulas involving the gamma function and factorials
  • Simple poles of the gamma function
• Euler’s limit formula: Second form of the gamma function
• Weierstrass’s infinite product: Third form of the gamma function
  • Proof of convergence for the Euler-Mascheroni constant
• 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 $\psi_{0} := \Gamma’ / \Gamma$
  • Derivative of the gamma function at 1 $\Gamma’ (1) = - \gamma$

Beta Function

• Beta function $B$
  • Trigonometric representation of the beta function
  • Relationship between the beta and gamma functions
  • Integral representation of the beta function
• Euler integrals: Beta and gamma functions

Riemann Zeta Function

• Riemann zeta function $\zeta$
  • Laurent expansion of the Riemann zeta function
• 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
  • Analytic proof of $1+1+1+1+1+⋯=-1/2$
  • Analytic proof of $1+2+3+4+5+⋯=-1/12$

Special Functions

• What are special functions?
• Airy function
• Rodrigues’ formula
• Laguerre polynomials
  • Rodrigues’ formula for Laguerre polynomials

Bessel Functions

• Bessel function $I_{\nu}$
  • Orthogonality
  • Recurrence relations
• 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

• Legendre polynomials
  • Orthogonality
    • Orthogonality with polynomials of lower order
  • Recurrence relations
  • Generating function
  • Rodrigues’ formula
• Associated Legendre polynomials
  • Orthogonality
  • Negative orders

Hermite Polynomials

• Hermite polynomials
  • Orthogonality
  • Recurrence relations
  • Generating function
  • Rodrigues’ formula
• Hermite functions
  • Operator solution of the differential equation satisfied by Hermite functions

Miscellaneous

• Why ‘implicit function’ is a mistranslation
• The history of the Dirac delta function and why Dirac used the delta function