## WeBWorK Problems

by Joel Trussell -
Number of replies: 0
I'm writing a problem on Laplace transforms and have asked the student to give the domain of convergence for a problem. the answer should be Re(s)>0
where I have declared s as a complex value
Context("Inequalities");
Context()->variables->are(
s=>"Complex"
);
$answerf = Inequality("Re(s)>0"); I get the error Function 'Re' is not allowed in this context The entire code is given below DOCUMENT(); loadMacros( "PGstandard.pl", "MathObjects.pl", "AnswerFormatHelp.pl", "PGunion.pl", "parserFunction.pl", "answerHints.pl", "parserAssignment.pl", "parserFormulaUpToConstant.pl", "contextInequalities.pl" ); TEXT(beginproblem()); ############################# # Setup1 Context("Complex"); Context()->variables->are( s=>"Complex", t=>"Real", dt =>"Real" ); Context()->functions->add( step => { class => 'Parser::Legacy::Numeric', perl => 'Parser::Legacy::Numeric::do_step' }, ); parserFunction("u(t)" => "step(t)");$a = random(1,9,1);

$ft = Formula("$a");

$answera = Compute("$ft * e^{-s*t} * dt");
$answerb = Compute("0");$answerc = Compute("INFINITY");
$answerd = Compute("$a/(-s) * e^{-s*t}");
$answere = Compute("$a/s");

$reals= Re(Compute($a));

#############################
# Main text1

Context()->texStrings;
BEGIN_TEXT
The Laplace transform of the function $$f(t)$$ is defined by the integral
$F(s) = \int_0^\infty f(t) e^{-st}dt$

where $$s = \alpha + j\omega$$ is a complex number. All of our functions will be defined only on the non-negative domain, $$[0,\infty]$$. We will usually emphasize this by including the step function, as $$f(t) = cos(2\pi t) u(t)$$. In any case, the Laplace integral is always defined with a lower bound great than or equal to zero.
$PAR Note that the Laplace transform, $$F(s)$$, is a function of $$s$$. We must make sure that the function is well-defined so the function actually exists. Let's take a look at an example to show how this works.$PAR
Consider the function $$f(t) = u(t)$$, the unit step function. The transform is defined by
$F(s) = \int_0^\infty f(t) e^{-st}dt = \int_0^\infty e^{-st}dt$

since $$u(t)=1$$, for all $$t \geq 0$$. This is an easy integral to compute, as
$F(s) = \frac{ e^{-st}}{-s}|_0^\infty$

This may seem straightforward, but we need to check to see if the function is defined at the limits of the integral. At the lower limit, it is easy
$\frac{ e^{-st}}{-s}|_0 = \frac{ e^{-s0}}{-s} = \frac{e^{-s0}}{-s} = \frac{-1}{s}$

but at the upper limit the funciton is not so clear. We have to take the limit as $$t \longrightarrow \infty$$. This limit depends on the value of $$s = \alpha + j\omega$$. We know from Euler's formula that
$e^{-st} = e^{(\alpha + j\omega)t} = e^{\alpha t }e^{ j\omega t}.$

We know that the magnitude of $$e^{ j\omega t}. = 1$$, so we are concerned only with the behavior of the limit of the real part, $$\lim_{t\rightarrow \intfy} e^{\alpha t }$$
$BR But we know this limit exists only if $$\alpha > 0$$ and the limit is zero. This can also be stated as $$Re(s) > 0$$, or the real part of $$s$$ is greater than zero.$BR So the evaluation of integral yields
$F(s) = \frac{ e^{-st}}{-s}|_0^\infty = \frac{1}{s}$

With this computation complete, let's look at a minor modification.

\{ BeginList('OL', type=>'a') \}

$ITEM Set up an integral for finding the Laplace transform of $$f(t) = ft u(t)$$.$BR
$BR $$\displaystyle F(s) = {\mathcal L}\left\lbrace f(t) \right\rbrace = \int_A^B$$ \{ ans_rule(20) \} \{ AnswerFormatHelp("formulas") \}$BR
$BR where $$A =$$ \{ ans_rule(5) \} and $$B =$$ \{ ans_rule(5) \}.$ITEMSEP
$ITEM Find the antiderivative and limits corresponding to the previous part.$BR
$BR \{ ans_rule(20) \}$${\Huge |}_A^B$$$ITEMSEP
$ITEM Evaluate appropriate limits to compute the Laplace transform of $$f(t)$$:$BR
$BR $$F(s) = {\mathcal L}\left\lbrace f(t) \right\rbrace =$$ \{ ans_rule(20) \} END_TEXT Context()->normalStrings; ############################## # Answer evaluation1$showPartialCorrectAnswers = 1;

ANS( $answera->cmp() ->withPostFilter(AnswerHints( Formula("$ft * e^(-s*t)") => "Don't forget the differential dt"
))
);
ANS( $answerb->cmp() ); ANS($answerc->cmp() );
ANS( $answerd->cmp(upToConstant=>1) ); ANS($answere->cmp() );

##############################
# Setup2

Context("Inequalities");
Context()->variables->are(
s=>"Complex"
);
$answerf = Inequality("Re(s)>0"); ############################## # Main text2 Context()->texStrings; BEGIN_TEXT$ITEMSEP
$ITEM Where does the Laplace transform you found exist? In other words, what is the domain of $$F(s)$$?$BR
$BR \{ ans_rule(20) \} \{ AnswerFormatHelp("inequalities") \} \{ EndList('OL') \} END_TEXT Context()->normalStrings; ############################## # Answer evaluation2 ANS($answerf->cmp() );

ENDDOCUMENT();