![]() G := convert − signum x ⋅ a − x, ' piecewise ', xĭsolve &DifferentialD &DifferentialD x y x = g ⋅ y x 2, y x ![]() Y x = &lcub &ExponentialE − x 2 erf I x _C1 + &ExponentialE − x 2 _C2 x Įq := &DifferentialD &DifferentialD x y x = piecewise 1 Y x = &lcub &ExponentialE − &ExponentialE x _C1 x Įq := &DifferentialD 2 &DifferentialD x 2 y x + 2 x &DifferentialD &DifferentialD x y x + 2 y x = piecewise 0 Some examples are included in the sections that follow.įirst-order Linear with Piecewise (Nontrivial) CoefficientsĮq := &DifferentialD &DifferentialD x y x + piecewise x The type of equations that one can solve include all first-order methods using integration, Riccati, and higher-order methods including linear, Bernoulli, and Euler. We can solve differential equations with piecewise functions in the coefficients. Solving Differential Equations with Piecewise H := convert min x 2 − 2, x − 1, ' piecewise ' Thus, it can be converted to a new function g ( x ). ![]() However, series can do better when using piecewise. This produces an answer with a superfluous order term. The straightforward calculation of the series of f around x =0 can be computed by using the series command. To determine the highest order of continuity and the problem points, enter:ĭerivatives can be found, and piecewise functions are returned. Isdifferentiable newcubic, x, 3, ' badpoints ' This must be true for splines! However, when we check to see if it is C 3, we obtain For example, in the case of our previous spline function, newcubic, we have We can also determine the differentiability class of a piecewise continuous function. It turns out to be a well-behaved, non-piecewise function. ![]() For example,Ĭonvert 1 − x, ' piecewise 'Ĭonvert − signum x 1 − x, ' piecewise ' Other piecewise functions can also be converted to piecewise and be properly manipulated. Newcubic ≔ CurveFitting Spline 0, 1, 2, 3, 0, 1, 4, 3, x Normal convert heavyf, ' piecewise ' Heavyf := x Heaviside 1 + x − x − x 2 Heaviside x − 1 + x 2 Heaviside 1 + x + sin x − 1 Heaviside x − 1 x − 1ĭistributions can be converted back to piecewise functions. Note: This works because discont is able to determine the potential discontinuities of piecewise functions. Where Si(x) is the Sine integral function. Using the same function, f ( x ), find its piecewise derivative.įprime ≔ &DifferentialD &DifferentialD x f Examples of solving DEs will be illustrated later. Such functions can be plotted to determine their behavior.īesides evaluating limits, you can do operations such as computing derivatives, integrating, and solving differential equations with piecewise functions. The next several Maple command lines make use of the following piecewise function:į := piecewise x ≤ − 1, − x, x ≤ 1, x 2, 1 īecause of the division by zero, points such as x = 1 cannot be substituted. Every piece is specified by a Boolean condition followed by an expression. The piecewise function has a straightforward syntax. Maxima used directly might be of help, or perhaps using its integration as at this Maxima feature request or using its piecewise capabilities.This worksheet contains a number of examples of the use of the piecewise function. discrete ones in Numpy, but your use case might be a little trickier. You can certainly get some convolutions, e.g. I'm hoping someone more familiar with some of the numerical tools in Sage can help you with your underlying questions. See the piecewise tag here and questions under it, such as this one. Stock disclaimer: all piecewise material dates from before Sage had any true symbolic capabilities. It's not clear to me that there is an easy way to get around this, because there isn't an obvious way to turn the polynomial generator x into an exponential, but the code for convolution relies pretty heavily on having those polynomial generators. Also, the example (and yours) uses a polynomial variable, which of course (?) becomes non-polynomial once e^x is involved. Piecewise defined function with 1 parts, ] I can't fix this for you (yet?), but I see why this wasn't detected before - the example you reference uses f = Piecewise(])
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