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Ramp Function

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RampFunction

The ramp function is defined by

R(x)=xH(x)
(1)
=int_(-infty)^xH(x^')dx^'
(2)
=int_(-infty)^inftyH(x^')H(x-x^')dx^'
(3)
=H(x)*H(x),
(4)

where H(x) is the Heaviside step function and * denotes convolution.

It is implemented in the Wolfram Language as Ramp[x].

The derivative is

R^'(x)=H(x).
(5)

The Fourier transform of the ramp function is given by

F_x[R(x)](k)=int_(-infty)^inftye^(-2piikx)R(x)dx
(6)
=(idelta^'(k))/(4pi)-1/(4pi^2k^2),
(7)

where delta(x) is the delta function and delta^'(x) its derivative.

See also

Fourier Transform--Ramp Function, Heaviside Step Function, Rectangle Function, Sawtooth Wave, Sign, Square Wave

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Cite this as:

Weisstein, Eric W. "Ramp Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RampFunction.html

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