Ramp Function

Author

Eric W. Weisstein
March 26, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GeneralizedFunctions/RampFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RampFunction.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Identities

(∫_(-∞)^x)(UnitStepSeq[x'])dx'==x (UnitStepSeq[x])

v4.2.1

Integrate[UnitStep[x],{x,-∞,x}]

v5.0.1

Integrate[UnitStep[x],{x,-∞,x}]

v5.1

Integrate[UnitStep[x],{x,-∞,x}]

V7

Integrate[UnitStep[x],{x,-∞,x}]

Integrate[UnitStep[x],{x,-∞,x},Assumptions->x∈Reals]

Integrate[HeavisideTheta[x],{x,-∞,x}]

Integrate[HeavisideTheta[x],{x,-∞,x},Assumptions->x∈Reals]

(∫_(-∞)^∞)(UnitStepSeq[x']) (UnitStepSeq[x-x'])dx'==x (UnitStepSeq[x])

v4.2.1

Integrate[UnitStep[xx]UnitStep[x-xx],{xx,-∞,∞}]

v5.1

Integrate[UnitStep[xx]UnitStep[x-xx],{xx,-∞,∞}]

v6.0

Integrate[UnitStep[xx]UnitStep[x-xx],{xx,-∞,∞}]

V7

Integrate[UnitStep[xx]UnitStep[x-xx],{xx,-∞,∞}]

Integrate[HeavisideTheta[xx]UnitStep[x-xx],{xx,-∞,∞}]

Integrate[UnitStep[xx]HeavisideTheta[x-xx],{xx,-∞,∞}]

Integrate[HeavisideTheta[xx]HeavisideTheta[x-xx],{xx,-∞,∞}]

Integral

Integrate[UnitStep[x],x]

Integrate[HeavisideTheta[x],x]

Derivative

RampFunction[x]

V5.2

D[x UnitStep[x],x]

D[x HeavisideTheta[x],x]

V6

D[x UnitStep[x],x]

D[x HeavisideTheta[x],x]

V7

D[x UnitStep[x],x]

FullSimplify[%,x∈Integers]

FullSimplify[%,x>0]

D[x HeavisideTheta[x],x]

FullSimplify[%]

Convolution

Convolve[UnitStep[x],UnitStep[x],x,t]

Convolve[HeavisideTheta[x],HeavisideTheta[x],x,t]

Plot

Plot[x UnitStep[x],{x,-2,2},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,R[x]}]

Fourier Transform

V6?

FourierTransform[RampFunction[x],x,k,FourierParameters->{0,-2Pi}]

V7

FourierTransform[RampFunction[x],x,k,FourierParameters->{0,-2Pi}]

InverseFourierTransform[%,k,x,FourierParameters->{0,-2Pi}]

FunctionExpand[%]