Piecewise

Piecewise[{{val1,cond1},{val2,cond2},}]

represents a piecewise function with values vali in the regions defined by the conditions condi.

Piecewise[{{val1,cond1},},val]

uses default value val if none of the condi apply. The default for val is 0.

Details

  • The condi are typically inequalities such as .
  • The condi are evaluated in turn, until one of them is found to yield True.
  • If all preceding condi yield False, then the vali corresponding to the first condi that yields True is returned as the value of the piecewise function.
  • If any of the preceding condi do not literally yield False, the Piecewise function is returned in symbolic form.
  • Only those vali explicitly included in the returned form are evaluated.
  • Elements of the form {vali,False} are dropped, as are all elements after the first {vali,True}.
  • Piecewise[conds] automatically evaluates to Piecewise[conds,0].
  • Piecewise can be used in such functions as Integrate, Minimize, Reduce, DSolve, and Simplify, as well as their numeric analogs.
  • Piecewise[{{v1,c1},{v2,c2},}] can be input in the form
    v1c1
    v2c2
    . The piecewise operator can be entered as pw or \[Piecewise]. The grid of values and conditions can be constructed by first entering , then using and .
  • In StandardForm and TraditionalForm, Piecewise[{{v1,c1},{v2,c2},}] is normally output using a brace, as in
    v1c1
    v2c2
    .

Examples

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Basic Examples  (3)

Set up a piecewise function with different pieces below and above zero:

Find the derivative of a piecewise function:

Use pw to enter and and then TemplateBox[{ctrl, return}, Key1, BaseStyle -> {ExampleText, FontWeight -> Plain, FontFamily -> Source Sans Pro}] for each additional piecewise case:

Scope  (12)

Define a piecewise function:

Evaluate it at specific points:

Plot it:

Refine it under assumptions:

Automatic simplification of Piecewise functions:

Remove unreachable cases:

Remove False conditions:

Merge cases with the same values:

If values are not specified in a region, they are assumed to be zero:

This specifies that the default value should be 1:

Compute limits of piecewise functions:

Compute the limit in the direction of the positive imaginary axis:

Compute the series of a piecewise function:

Integrate a piecewise function:

Integration constants are chosen to make the result continuous:

Compute a definite integral of a piecewise function:

Laplace transform of a piecewise function:

Solve a piecewise differential equation:

Reduce a piecewise equation:

Integrating an implicitly piecewise integrand can give an explicit Piecewise result:

Symbolic minimization can give piecewise functions:

Applications  (1)

Compute the volume of an ellipsoid:

Properties & Relations  (11)

PiecewiseExpand converts nested piecewise functions into a single piecewise function:

Min, Max, UnitStep, and Clip are piecewise functions of real arguments:

Abs, Sign, and Arg are piecewise functions when their arguments are assumed to be real:

KroneckerDelta and DiscreteDelta are piecewise functions of complex arguments:

Boole is a piecewise function of a Boolean argument:

If, Which, and Switch can be interpreted as piecewise functions:

Convert Floor, Ceiling, Round, IntegerPart, and FractionalPart for finite ranges:

Convert Mod and Quotient when the number of cases is finite:

UnitBox and UnitTriangle are piecewise functions of real arguments:

Convert SquareWave, TriangleWave, and SawtoothWave for finite ranges:

BernsteinBasis and BSplineBasis are piecewise functions of real arguments:

Possible Issues  (1)

Derivatives are computed piece-by-piece, unless the function is univariate in a real variable:

To specify that is real, use inequalities in the first condition:

This function is discontinuous at :

Introduced in 2004
 (5.1)
 |
Updated in 2008
 (7.0)