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 : F → G {\displaystyle L:{\mathcal {F}}\to {\mathcal {G}}}which takes a function y ∈ F {\displaystyle y\in {\mathcal {F}}} to another function L [ y ] ∈ G {\displaystyle L[y]\in {\mathcal {G}}} . Here, F {\displaystyle {\mathcal {F}}} and G {\displaystyle {\mathcal {G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions.
| Expression | Curvedefinition | Variables | Description |
|---|---|---|---|
| Linear transformations | |||
| L [ y ] = y ( n ) {\displaystyle L[y]=y^{(n)}} | Derivative of nth order | ||
| L [ y ] = ∫ a t y d t {\displaystyle L[y]=\int _{a}^{t}y\,dt} | Cartesian | y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} | Integral, area |
| L [ y ] = y ∘ f {\displaystyle L[y]=y\circ f} | Composition operator | ||
| L [ y ] = y ∘ t + y ∘ − t 2 {\displaystyle L[y]={\frac {y\circ t+y\circ -t}{2}}} | Even component | ||
| L [ y ] = y ∘ t − y ∘ − t 2 {\displaystyle L[y]={\frac {y\circ t-y\circ -t}{2}}} | Odd component | ||
| L [ y ] = y ∘ ( t + 1 ) − y ∘ t = Δ y {\displaystyle L[y]=y\circ (t+1)-y\circ t=\Delta y} | Difference operator | ||
| L [ y ] = y ∘ ( t ) − y ∘ ( t − 1 ) = ∇ y {\displaystyle L[y]=y\circ (t)-y\circ (t-1)=\nabla y} | Backward difference (Nabla operator) | ||
| L [ y ] = ∑ y = Δ − 1 y {\displaystyle L[y]=\sum y=\Delta ^{-1}y} | Indefinite sum operator (inverse operator of difference) | ||
| L [ y ] = − ( p y ′ ) ′ + q y {\displaystyle L[y]=-(py')'+qy} | Sturm–Liouville operator | ||
| Non-linear transformations | |||
| F [ y ] = y [ − 1 ] {\displaystyle F[y]=y^{[-1]}} | Inverse function | ||
| F [ y ] = t y ′ [ − 1 ] − y ∘ y ′ [ − 1 ] {\displaystyle F[y]=t\,y'^{[-1]}-y\circ y'^{[-1]}} | Legendre transformation | ||
| F [ y ] = f ∘ y {\displaystyle F[y]=f\circ y} | Left composition | ||
| F [ y ] = ∏ y {\displaystyle F[y]=\prod y} | Indefinite product | ||
| F [ y ] = y ′ y {\displaystyle F[y]={\frac {y'}{y}}} | Logarithmic derivative | ||
| F [ y ] = t y ′ y {\displaystyle F[y]={\frac {ty'}{y}}} | Elasticity | ||
| F [ y ] = y ‴ y ′ − 3 2 ( y ″ y ′ ) 2 {\displaystyle F[y]={y''' \over y'}-{3 \over 2}\left({y'' \over y'}\right)^{2}} | Schwarzian derivative | ||
| F [ y ] = ∫ a t | y ′ | d t {\displaystyle F[y]=\int _{a}^{t}|y'|\,dt} | Total variation | ||
| F [ y ] = 1 t − a ∫ a t y d t {\displaystyle F[y]={\frac {1}{t-a}}\int _{a}^{t}y\,dt} | Arithmetic mean | ||
| F [ y ] = exp ( 1 t − a ∫ a t ln y d t ) {\displaystyle F[y]=\exp \left({\frac {1}{t-a}}\int _{a}^{t}\ln y\,dt\right)} | Geometric mean | ||
| F [ y ] = − y y ′ {\displaystyle F[y]=-{\frac {y}{y'}}} | Cartesian | y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} | Subtangent |
| F [ x , y ] = − y x ′ y ′ {\displaystyle F[x,y]=-{\frac {yx'}{y'}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | |
| F [ r ] = − r 2 r ′ {\displaystyle F[r]=-{\frac {r^{2}}{r'}}} | Polar | r = r ( ϕ ) {\displaystyle r=r(\phi )} ϕ = t {\displaystyle \phi =t} | |
| F [ r ] = 1 2 ∫ a t r 2 d t {\displaystyle F[r]={\frac {1}{2}}\int _{a}^{t}r^{2}dt} | Polar | r = r ( ϕ ) {\displaystyle r=r(\phi )} ϕ = t {\displaystyle \phi =t} | Sector area |
| F [ y ] = ∫ a t 1 + y ′ 2 d t {\displaystyle F[y]=\int _{a}^{t}{\sqrt {1+y'^{2}}}\,dt} | Cartesian | y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} | Arc length |
| F [ x , y ] = ∫ a t x ′ 2 + y ′ 2 d t {\displaystyle F[x,y]=\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | |
| F [ r ] = ∫ a t r 2 + r ′ 2 d t {\displaystyle F[r]=\int _{a}^{t}{\sqrt {r^{2}+r'^{2}}}\,dt} | Polar | r = r ( ϕ ) {\displaystyle r=r(\phi )} ϕ = t {\displaystyle \phi =t} | |
| F [ y ] = ∫ a t y ″ 3 d t {\displaystyle F[y]=\int _{a}^{t}{\sqrt[{3}]{y''}}\,dt} | Cartesian | y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} | Affine arc length |
| F [ x , y ] = ∫ a t x ′ y ″ − x ″ y ′ 3 d t {\displaystyle F[x,y]=\int _{a}^{t}{\sqrt[{3}]{x'y''-x''y'}}\,dt} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | |
| F [ x , y , z ] = ∫ a t z ‴ ( x ′ y ″ − y ′ x ″ ) + z ″ ( x ‴ y ′ − x ′ y ‴ ) + z ′ ( x ″ y ‴ − x ‴ y ″ ) 3 d t {\displaystyle F[x,y,z]=\int _{a}^{t}{\sqrt[{3}]{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}}dt} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} z = z ( t ) {\displaystyle z=z(t)} | |
| F [ y ] = y ″ ( 1 + y ′ 2 ) 3 / 2 {\displaystyle F[y]={\frac {y''}{(1+y'^{2})^{3/2}}}} | Cartesian | y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} | Curvature |
| F [ x , y ] = x ′ y ″ − y ′ x ″ ( x ′ 2 + y ′ 2 ) 3 / 2 {\displaystyle F[x,y]={\frac {x'y''-y'x''}{(x'^{2}+y'^{2})^{3/2}}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | |
| F [ r ] = r 2 + 2 r ′ 2 − r r ″ ( r 2 + r ′ 2 ) 3 / 2 {\displaystyle F[r]={\frac {r^{2}+2r'^{2}-rr''}{(r^{2}+r'^{2})^{3/2}}}} | Polar | r = r ( ϕ ) {\displaystyle r=r(\phi )} ϕ = t {\displaystyle \phi =t} | |
| F [ x , y , z ] = ( z ″ y ′ − z ′ y ″ ) 2 + ( x ″ z ′ − z ″ x ′ ) 2 + ( y ″ x ′ − x ″ y ′ ) 2 ( x ′ 2 + y ′ 2 + z ′ 2 ) 3 / 2 {\displaystyle 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}}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} z = z ( t ) {\displaystyle z=z(t)} | |
| F [ y ] = 1 3 y ⁗ ( y ″ ) 5 / 3 − 5 9 y ‴ 2 ( y ″ ) 8 / 3 {\displaystyle F[y]={\frac {1}{3}}{\frac {y''''}{(y'')^{5/3}}}-{\frac {5}{9}}{\frac {y'''^{2}}{(y'')^{8/3}}}} | Cartesian | y = y ( x ) {\displaystyle y=y(x)} x = t {\displaystyle x=t} | Affine curvature |
| F [ x , y ] = x ″ y ‴ − x ‴ y ″ ( x ′ y ″ − x ″ y ′ ) 5 / 3 − 1 2 [ 1 ( x ′ y ″ − x ″ y ′ ) 2 / 3 ] ″ {\displaystyle F[x,y]={\frac {x''y'''-x'''y''}{(x'y''-x''y')^{5/3}}}-{\frac {1}{2}}\left[{\frac {1}{(x'y''-x''y')^{2/3}}}\right]''} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | |
| F [ x , y , z ] = 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 ) {\displaystyle 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})}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} z = z ( t ) {\displaystyle z=z(t)} | Torsion of curves |
| X [ x , y ] = y ′ y x ′ − x y ′ {\displaystyle X[x,y]={\frac {y'}{yx'-xy'}}} Y [ x , y ] = x ′ x y ′ − y x ′ {\displaystyle Y[x,y]={\frac {x'}{xy'-yx'}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | Dual curve(tangent coordinates) |
| X [ x , y ] = x + a y ′ x ′ 2 + y ′ 2 {\displaystyle X[x,y]=x+{\frac {ay'}{\sqrt {x'^{2}+y'^{2}}}}} Y [ x , y ] = y − a x ′ x ′ 2 + y ′ 2 {\displaystyle Y[x,y]=y-{\frac {ax'}{\sqrt {x'^{2}+y'^{2}}}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | Parallel curve |
| X [ x , y ] = x + y ′ x ′ 2 + y ′ 2 x ″ y ′ − y ″ x ′ {\displaystyle X[x,y]=x+y'{\frac {x'^{2}+y'^{2}}{x''y'-y''x'}}} Y [ x , y ] = y + x ′ x ′ 2 + y ′ 2 y ″ x ′ − x ″ y ′ {\displaystyle Y[x,y]=y+x'{\frac {x'^{2}+y'^{2}}{y''x'-x''y'}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | Evolute |
| F [ r ] = t ( r ′ ∘ r [ − 1 ] ) {\displaystyle F[r]=t(r'\circ r^{[-1]})} | Intrinsic | r = r ( s ) {\displaystyle r=r(s)} s = t {\displaystyle s=t} | |
| X [ x , y ] = x − x ′ ∫ a t x ′ 2 + y ′ 2 d t x ′ 2 + y ′ 2 {\displaystyle X[x,y]=x-{\frac {x'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}} Y [ x , y ] = y − y ′ ∫ a t x ′ 2 + y ′ 2 d t x ′ 2 + y ′ 2 {\displaystyle Y[x,y]=y-{\frac {y'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | Involute |
| X [ x , y ] = ( x y ′ − y x ′ ) y ′ x ′ 2 + y ′ 2 {\displaystyle X[x,y]={\frac {(xy'-yx')y'}{x'^{2}+y'^{2}}}} Y [ x , y ] = ( y x ′ − x y ′ ) x ′ x ′ 2 + y ′ 2 {\displaystyle Y[x,y]={\frac {(yx'-xy')x'}{x'^{2}+y'^{2}}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | Pedal curve with pedal point (0;0) |
| X [ x , y ] = ( x ′ 2 − y ′ 2 ) y ′ + 2 x y x ′ x y ′ − y x ′ {\displaystyle X[x,y]={\frac {(x'^{2}-y'^{2})y'+2xyx'}{xy'-yx'}}} Y [ x , y ] = ( x ′ 2 − y ′ 2 ) x ′ + 2 x y y ′ x y ′ − y x ′ {\displaystyle Y[x,y]={\frac {(x'^{2}-y'^{2})x'+2xyy'}{xy'-yx'}}} | ParametricCartesian | x = x ( t ) {\displaystyle x=x(t)} y = y ( t ) {\displaystyle y=y(t)} | Negative pedal curve with pedal point (0;0) |
| X [ y ] = ∫ a t cos [ ∫ a t 1 y d t ] d t {\displaystyle X[y]=\int _{a}^{t}\cos \left[\int _{a}^{t}{\frac {1}{y}}\,dt\right]dt} Y [ y ] = ∫ a t sin [ ∫ a t 1 y d t ] d t {\displaystyle Y[y]=\int _{a}^{t}\sin \left[\int _{a}^{t}{\frac {1}{y}}\,dt\right]dt} | Intrinsic | y = r ( s ) {\displaystyle y=r(s)} s = t {\displaystyle s=t} | Intrinsic toCartesiantransformation |
| Metric functionals | |||
| F [ y ] = ‖ y ‖ = ∫ E y 2 d t {\displaystyle F[y]=\|y\|={\sqrt {\int _{E}y^{2}\,dt}}} | Norm | ||
| F [ x , y ] = ∫ E x y d t {\displaystyle F[x,y]=\int _{E}xy\,dt} | Inner product | ||
| F [ x , y ] = arccos [ ∫ E x y d t ∫ E x 2 d t ∫ E y 2 d t ] {\displaystyle 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 = ∫ E x ( s ) y ( t − s ) d s {\displaystyle F[x,y]=x*y=\int _{E}x(s)y(t-s)\,ds} | Convolution | ||
| F [ y ] = ∫ E y ln y d t {\displaystyle F[y]=\int _{E}y\ln y\,dt} | Differential entropy | ||
| F [ y ] = ∫ E y t d t {\displaystyle F[y]=\int _{E}yt\,dt} | Expected value | ||
| F [ y ] = ∫ E ( t − ∫ E y t d t ) 2 y d t {\displaystyle F[y]=\int _{E}\left(t-\int _{E}yt\,dt\right)^{2}y\,dt} | Variance | ||