BUILT-IN MATHEMATICA SYMBOL

Piecewise

represents a piecewise function with values

in the regions defined by the conditions

.

Piecewise

uses default value val if none of the

apply. The default for val is

.

MORE INFORMATION

The are typically inequalities such as .

The are evaluated in turn, until one of them is found to yield True.

If all preceding yield False, then the corresponding to the first that yields True is returned as the value of the piecewise function.

If any of the preceding do not literally yield False, the Piecewise function is returned in symbolic form.

Only those explicitly included in the returned form are evaluated.

Elements of the form {val_i, False} are dropped, as are all elements after the first {val_i, True}.

Piecewise[conds] automatically evaluates to Piecewise.

Piecewise can be used in such functions as Integrate, Minimize, Reduce, DSolve, and Simplify, as well as their numeric analogs.

Piecewise can be input in the form . The piecewise operator can be entered as Esc pw Esc or \[Piecewise]. The grid of values and conditions can be constructed by first entering Ctrl+Comma, then using Ctrl+

and Ctrl+Comma.

In StandardForm and TraditionalForm, Piecewise is normally output using a brace, as in .

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 Esc pw Esc to enter

and Ctrl+Comma and then

for each additional piecewise case:

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

In[1]:=

Out[1]=

Find the derivative of a piecewise function:

In[1]:=

Out[1]=

Use Esc pw Esc to enter

and Ctrl+Comma and then

for each additional piecewise case:

In[1]:=

Out[1]//InputForm=

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:

TUTORIALS

Piecewise Functions

Conditionals

MORE ABOUT

Conditionals

Inequalities

Numerical Functions

New in 5.1