List of operators

From Wikipedia, the free encyclopedia

In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators.

In the following L is an operator

L:\mathcal{F}\to\mathcal{G}

which takes a function y\in\mathcal{F} to another function L[y]\in\mathcal{G}. Here, \mathcal{F} and \mathcal{G} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions.

Expression Curve
definition		 Variables Description
Linear transformations
L[y]=y^{(n)} \ Derivative of n-th order
L[y]=\int_a^t y \,dt Cartesian y = y(x)
x = t		 Integral, area
L[y]=y\circ f Composition operator
L[y]=\frac{y\circ t+y\circ -t}{2} Even component
L[y]=\frac{y\circ t-y\circ -t}{2} Odd component
L[y] =-(py')'+qy \, Sturm-Liouville operator
L[y]=\int_0^\infty y(s)\exp{(-ts)}\,ds Laplace transform
L[y]= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty y(s) \exp{(- its)}\,ds Fourier transform
L[y] =\int_0^{\infty} \frac{s^t y(s)}{s}\,ds. Mellin transform
L[y]=2\int_t^\infty \frac{y(s)s\,ds}{\sqrt{s^2-t^2}}. Abel transform
L[y]=-\frac{1}{\pi}\int_t^\infty \frac{y'(s)\,ds}{\sqrt{s^2-t^2}}. Inverse Abel transform
L[y]= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty y(s) \cos ts \sin ts \,ds Hartley transform
Non-linear transformations
F[y]=y^{-1}=\mbox{inv } y \ Inverse function
F[y]=t\,\mbox{inv }y' - y\circ \mbox{inv }y' Legendre transformation
F[y]=f\circ y Left composition
F[y]=\frac{y'}{y} Logarithmic derivative
F[y]=\int_a^t |y'| \,dt Total variation
F[y]=\frac{1}{t-a}\int_a^t y\,dt Mean value
F[y]=\exp \left( \frac{1}{t-a}\int_a^t \ln y\,dt \right) Geometric mean value
F[y]= -\frac{y}{y'} Cartesian y = y(x)
x = t		 Subtangent
F[x,y]= -\frac{yx'}{y'} Parametric
Cartesian		 x = x(t)
y = y(t)
F[y]= -\frac{y^2}{y'} Polar y = r(φ)
φ = t
F[y]=\frac{1}{2}\int_a^t y^2 dt Polar y = r(φ)
φ = t		 Area
F[y]= \int_a^t \sqrt { 1 + y'^2 }\, dt Cartesian y = y(x)
x = t		 Arc length
F[x,y]= \int_a^t \sqrt { x'^2 + y'^2 }\, dt Parametric
Cartesian		 x = x(t)
y = y(t)
F[y]= \int_a^t \sqrt { y^2 + y'^2 }\, dt Polar y = r(φ)
φ = t
F[y]=\frac{y''}{(1+y'^2)^{3/2}} Cartesian y = y(x)
x = t		 Curvature
F[x,y]= \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}} Parametric
Cartesian		 x = x(t)
y = y(t)
F[y]=\frac{y^2+2y'^2-yy''}{(y^2+y'^2)^{3/2}} Polar y = r(φ)
φ = t
F[x,y,z]=\frac{\sqrt{(z''y'-z'y'')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}} Parametric
Cartesian		 x = x(t)
y = y(t)
z = z(t)
F[x,y]=\left| \frac{x''y'''-x'''y''}{(x'y''-x''y')^{5/2}}-\frac{1}{2}\left[\frac{1}{(x'y''-x''y')}\right]''\right| Parametric
Cartesian		 x = x(t)
y = y(t)		 Affine curvature
F[x,y,z]=\frac{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)} Parametric
Cartesian		 x = x(t)
y = y(t)
z = z(t)		 Torsion of curves
X[x,y]=\frac{y'}{yx'-xy'}

Y[x,y]=\frac{x'}{xy'-yx'}		 Parametric
Cartesian		 x = x(t)
y = y(t)		 Dual curve
(tangent coordinates)
X[x,y]=x+\frac{ay'}{\sqrt {x'^2+y'^2}}

Y[x,y]=y-\frac{ax'}{\sqrt {x'^2+y'^2}}		 Parametric
Cartesian		 x = x(t)
y = y(t)		 Parallel curve
X[y]=t-\frac{1+y'^2}{y''}

Y[y]=y+\frac{1+y'^2}{y''}		 Cartesian y = y(x)
x = t		 Evolute
X[x,y]=x+y'\frac{x'^2+y'^2}{x''y'-y''x'}

Y[x,y]=y+x'\frac{x'^2+y'^2}{y''x'-x''y'}		 Parametric
Cartesian		 x = x(t)
y = y(t)
F[y]=\frac{yy'}{(\mbox{inv }y)'} Intrinsic y = r(s)
s = t
X[x,y]=x-\frac{x'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}

Y[x,y]=y-\frac{y'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}		 Parametric
Cartesian		 x = x(t)
y = y(t)		 Involute
X[x,y]=\frac{(xy'-yx')y'}{x'^2 + y'^2}

Y[x,y]=\frac{(yx'-xy')x'}{x'^2 + y'^2}		 Parametric
Cartesian		 x = x(t)
y = y(t)		 Pedal curve with pedal point (0;0)
X[x,y]=\frac{(x'^2-y'^2)y'+2xyx'}{xy'-yx'}

Y[x,y]=\frac{(x'^2-y'^2)x'+2xyy'}{xy'-yx'}		 Parametric
Cartesian		 x = x(t)
y = y(t)		 Negative pedal curve with pedal point (0;0)
X[y] = \int_a^t \cos \left[\int_a^t \frac{1}{y} \,dt\right] dt

Y[y] = \int_a^t \sin \left[\int_a^t \frac{1}{y} \,dt\right] dt		 Intrinsic y = r(s)
s = t		 Intrinsic to
Cartesian
transformation
Metric functionals
F[y]=||y||=\sqrt{\int_E y^2 \, dt} Norm
F[x,y]=\int_E xy \, dt Inner product
F[x,y]=\arccos \left[\frac{\int_E xy \, dt}{\sqrt{\int_E x^2 \, dt}\sqrt{\int_E y^2 \, dt}}\right] Fubini-Study metric
(inner angle)
Distribution functionals
F[x,y] = x * y = \int_E x(s) y(t - s)\, ds Convolution
F[y] = \int_E y \ln y \, dy Differential entropy
F[y] = \int_E yt\,dt Expected value
F[y] = \int_E (t-\int_E yt\,dt)^2y\,dt Variance

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