General analytic functions

1b1ec5

f\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right)

TeX:

f\!\left(z + x\right) = \sum_{k=0}^{\infty} \frac{{f}^{(k)}(z)}{k !} {x}^{k}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, \left|x\right|\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("1b1ec5"),
    Formula(Equal(f(Add(z, x)), Sum(Mul(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k)), Pow(x, k)), For(k, 0, Infinity)))),
    Variables(f, z, x),
    Assumptions(And(Element(z, CC), Element(x, CC), IsHolomorphic(f(t), ForElement(t, ClosedDisk(z, Abs(x)))))))

b6582a

\left|\frac{{f}^{(k)}(z)}{k !}\right| \le \frac{C}{{R}^{k}}\; \text{ where } C = \mathop{\operatorname{sup}}\limits_{t \in \mathbb{C},\,\left|t - z\right| = R} \left|f(t)\right|

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; R > 0 \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, R\right)

TeX:

\left|\frac{{f}^{(k)}(z)}{k !}\right| \le \frac{C}{{R}^{k}}\; \text{ where } C = \mathop{\operatorname{sup}}\limits_{t \in \mathbb{C},\,\left|t - z\right| = R} \left|f(t)\right|

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; R > 0 \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, R\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate

Source code for this entry:

Entry(ID("b6582a"),
    Formula(Where(LessEqual(Abs(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k))), Div(C, Pow(R, k))), Equal(C, Supremum(Abs(f(t)), ForElement(t, CC), Equal(Abs(Sub(t, z)), R))))),
    Variables(f, z, k, R),
    Assumptions(And(Element(z, CC), Element(k, ZZGreaterEqual(0)), Element(R, RR), Greater(R, 0), IsHolomorphic(f(t), ForElement(t, Subset(ClosedDisk(z, R)))))))

78bb08

\left|f\!\left(z + x\right) - \sum_{k=0}^{N - 1} \frac{{f}^{(k)}(z)}{k !} {x}^{k}\right| \le \frac{C {D}^{N}}{1 - D}\; \text{ where } C = \mathop{\operatorname{sup}}\limits_{t \in \mathbb{C},\,\left|t - z\right| = R} \left|f(t)\right|,\;D = \frac{\left|x\right|}{R}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| < R \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, R\right)

TeX:

\left|f\!\left(z + x\right) - \sum_{k=0}^{N - 1} \frac{{f}^{(k)}(z)}{k !} {x}^{k}\right| \le \frac{C {D}^{N}}{1 - D}\; \text{ where } C = \mathop{\operatorname{sup}}\limits_{t \in \mathbb{C},\,\left|t - z\right| = R} \left|f(t)\right|,\;D = \frac{\left|x\right|}{R}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; R \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|x\right| < R \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \operatorname{ClosedDisk}\!\left(z, R\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Sum`	$\sum_{n} f(n)$	Sum
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Factorial`	$n !$	Factorial
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate

Source code for this entry:

Entry(ID("78bb08"),
    Formula(Where(LessEqual(Abs(Sub(f(Add(z, x)), Sum(Mul(Div(ComplexDerivative(f(z), For(z, z, k)), Factorial(k)), Pow(x, k)), For(k, 0, Sub(N, 1))))), Div(Mul(C, Pow(D, N)), Sub(1, D))), Equal(C, Supremum(Abs(f(t)), For(t), And(Element(t, CC), Equal(Abs(Sub(t, z)), R)))), Equal(D, Div(Abs(x), R)))),
    Variables(f, z, x, N, R),
    Assumptions(And(Element(z, CC), Element(x, CC), Element(N, ZZGreaterEqual(1)), Element(R, RR), Less(Abs(x), R), IsHolomorphic(f(t), ForElement(t, Subset(ClosedDisk(z, R)))))))

545987

\left|\int_{a}^{b} f(t) \, dt - \frac{b - a}{2} \sum_{k=1}^{n} w_{n,k} f\!\left(\frac{b - a}{2} x_{n,k} + \frac{a + b}{2}\right)\right| \le \frac{\left|b - a\right|}{2} \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f\!\left(\frac{b - a}{2} t + \frac{a + b}{2}\right)\right|

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \rho \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \rho > 1 \;\mathbin{\operatorname{and}}\; f(z) \text{ is holomorphic on } z \in \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right)

References:

L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831

TeX:

\left|\int_{a}^{b} f(t) \, dt - \frac{b - a}{2} \sum_{k=1}^{n} w_{n,k} f\!\left(\frac{b - a}{2} x_{n,k} + \frac{a + b}{2}\right)\right| \le \frac{\left|b - a\right|}{2} \frac{64 M}{15 \left(1 - {\rho}^{-2}\right) {\rho}^{2 n}}\; \text{ where } M = \mathop{\operatorname{sup}}\limits_{t \in \mathcal{E}_{\rho}} \left|f\!\left(\frac{b - a}{2} t + \frac{a + b}{2}\right)\right|

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \rho \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \rho > 1 \;\mathbin{\operatorname{and}}\; f(z) \text{ is holomorphic on } z \in \operatorname{InteriorClosure}\!\left(\mathcal{E}_{\rho}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Sum`	$\sum_{n} f(n)$	Sum
`GaussLegendreWeight`	$w_{n,k}$	Gauss-Legendre quadrature weight
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`Pow`	${a}^{b}$	Power
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum of a set or function
`BernsteinEllipse`	$\mathcal{E}_{\rho}$	Bernstein ellipse with foci -1,+1 and semi-axis sum rho
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate

Source code for this entry:

Entry(ID("545987"),
    Formula(Where(LessEqual(Abs(Sub(Integral(f(t), For(t, a, b)), Mul(Div(Sub(b, a), 2), Sum(Mul(GaussLegendreWeight(n, k), f(Add(Mul(Div(Sub(b, a), 2), LegendrePolynomialZero(n, k)), Div(Add(a, b), 2)))), For(k, 1, n))))), Mul(Div(Abs(Sub(b, a)), 2), Div(Mul(64, M), Mul(Mul(15, Sub(1, Pow(rho, -2))), Pow(rho, Mul(2, n)))))), Equal(M, Supremum(Abs(f(Add(Mul(Div(Sub(b, a), 2), t), Div(Add(a, b), 2)))), ForElement(t, BernsteinEllipse(rho)))))),
    Variables(f, a, b, n, rho),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(n, ZZGreaterEqual(1)), Element(rho, RR), Greater(rho, 1), IsHolomorphic(f(z), ForElement(z, Subset(InteriorClosure(BernsteinEllipse(rho))))))),
    References("L. N. Trefethen, Is Gauss Quadrature Better than Clenshaw-Curtis? SIAM Rev., 50(1), 67-87. DOI:10.1137/060659831"))

ce2272

\sum_{k=N}^{U} f(k) = \int_{N}^{U} f(t) \, dt + \frac{f(N) + f(U)}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \left({f}^{(2 k - 1)}(U) - {f}^{(2 k - 1)}(N)\right) + \int_{N}^{U} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} {f}^{(2 M)}(t) \, dt

Assumptions:

N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; U \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \le U \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \left[N, U\right]

TeX:

\sum_{k=N}^{U} f(k) = \int_{N}^{U} f(t) \, dt + \frac{f(N) + f(U)}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \left({f}^{(2 k - 1)}(U) - {f}^{(2 k - 1)}(N)\right) + \int_{N}^{U} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} {f}^{(2 M)}(t) \, dt

N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; U \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \le U \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \left[N, U\right]

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`BernoulliPolynomial`	$B_{n}\!\left(z\right)$	Bernoulli polynomial
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("ce2272"),
    Formula(Equal(Sum(f(k), For(k, N, U)), Add(Add(Integral(f(t), For(t, N, U)), Add(Div(Add(f(N), f(U)), 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Sub(ComplexDerivative(f(t), For(t, U, Sub(Mul(2, k), 1))), ComplexDerivative(f(t), For(t, N, Sub(Mul(2, k), 1))))), For(k, 1, M)))), Integral(Mul(Div(BernoulliPolynomial(Mul(2, M), Sub(t, Floor(t))), Factorial(Mul(2, M))), ComplexDerivative(f(t), For(t, t, Mul(2, M)))), For(t, N, U))))),
    Variables(f, N, U, M),
    Assumptions(And(Element(N, ZZ), Element(U, ZZ), LessEqual(N, U), Element(M, ZZGreaterEqual(1)), IsHolomorphic(f(t), ForElement(t, Subset(ClosedInterval(N, U)))))))

af2d4b

\left|\sum_{k=N}^{U} f(k) - \left(\int_{N}^{U} f(t) \, dt + \frac{f(N) + f(U)}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \left({f}^{(2 k - 1)}(U) - {f}^{(2 k - 1)}(N)\right)\right)\right| \le \frac{4}{{\left(2 \pi\right)}^{2 M}} \int_{N}^{U} \left|{f}^{(2 M)}(t)\right| \, dt

Assumptions:

N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; U \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \le U \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \left[N, U\right]

TeX:

\left|\sum_{k=N}^{U} f(k) - \left(\int_{N}^{U} f(t) \, dt + \frac{f(N) + f(U)}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \left({f}^{(2 k - 1)}(U) - {f}^{(2 k - 1)}(N)\right)\right)\right| \le \frac{4}{{\left(2 \pi\right)}^{2 M}} \int_{N}^{U} \left|{f}^{(2 M)}(t)\right| \, dt

N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; U \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N \le U \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; f(t) \text{ is holomorphic on } t \in \left[N, U\right]

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Sum`	$\sum_{n} f(n)$	Sum
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`BernoulliB`	$B_{n}$	Bernoulli number
`Factorial`	$n !$	Factorial
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`ClosedInterval`	$\left[a, b\right]$	Closed interval

Source code for this entry:

Entry(ID("af2d4b"),
    Formula(LessEqual(Abs(Sub(Sum(f(k), For(k, N, U)), Parentheses(Add(Integral(f(t), For(t, N, U)), Add(Div(Add(f(N), f(U)), 2), Sum(Mul(Div(BernoulliB(Mul(2, k)), Factorial(Mul(2, k))), Sub(ComplexDerivative(f(t), For(t, U, Sub(Mul(2, k), 1))), ComplexDerivative(f(t), For(t, N, Sub(Mul(2, k), 1))))), For(k, 1, M))))))), Mul(Div(4, Pow(Mul(2, Pi), Mul(2, M))), Integral(Abs(ComplexDerivative(f(t), For(t, t, Mul(2, M)))), For(t, N, U))))),
    Variables(f, N, U, M),
    Assumptions(And(Element(N, ZZ), Element(U, ZZ), LessEqual(N, U), Element(M, ZZGreaterEqual(1)), IsHolomorphic(f(t), ForElement(t, Subset(ClosedInterval(N, U)))))))

Taylor series

Quadrature

Euler-Maclaurin formula