Unit Step Functions

Here is a Maple definition of the unit step function, , whose discontinuity occurs at . Note that the Maple implementation puts the location of the jump, , as the first argument of the function and the independent variable as the second argument.

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> u := ( a, t ) -> piecewise( t<a, 0, t>=a, 1 ):

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In order to better understand step functions and how they can be used in problems with piecewise-defined forcing functions, let's consider a few examples. First, consider the unit step function with jump at :

> plot( u(2,t), t=0..5, view=[0..5,-0.2..2], discont=true );

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Multplying a unit step function by another function has the effect of "turning off" the function until the step function "turns on" (at ).

(Compare the following plot with the plot of .)

> plot( t*u(2,t), t=0..5, discont=true );

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A summation of terms involving step functions can be understood by simplifying the expression on each subinterval on which a different collection of step functions is activated. In this case we have the function that is zero for < 1, t for 1 <= < 3, and 3 for >= 3.

> plot( t*u(1,t) + (3-t)*u(3,t), t=0..5, discont=true );

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And, finally, let's compare the two functions and .

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> f1 := sin(2*t)*u(1,t);

> f2 := sin(2*(t-1))*u(1,t);

> plot( f1, t=0..3 );

> plot( f2, t=0..3 );

Notice how the graph of is the graph of that is "turned off" until . On the other hand, the graph of looks like the sine function shifted one unit to the right. It is this latter form that is most convenient for use with Laplace transforms (see Table 8.2, p. 594).

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