With

With[{x=x0,y=y0,…},expr]

specifies that all occurrences of the symbols x, y, … in expr should be replaced by x0, y0, ….

Details

  • With allows you to define local constants.
  • With replaces symbols in expr only when they do not occur as local variables inside scoping constructs.
  • With[{x:=x0,…},expr] inserts the unevaluated form x0 into expr.
  • You can use With[{vars},body/;cond] as the right‐hand side of a transformation rule with a condition attached. »
  • With has attribute HoldAll.
  • With constructs can be nested in any way, with inner variables being renamed if necessary.
  • With is a scoping construct that implements read‐only lexical variables.

Examples

open allclose all

Basic Examples  (3)

Evaluate an expression with x locally set to 7:

Locally set both x and y:

With works even without evaluation:

Scope  (4)

Use With to insert values into held expressions:

Use := to insert the unevaluated form of an expression into the result:

Compare with the following:

The variable names can be the same:

Use a constant for a value that is needed more than once:

Applications  (1)

With allows inserting values into unevaluated expressions:

Properties & Relations  (5)

Using x:=x0 inserts the unevaluated form of x0, which then evaluates based on its surroundings:

Using x=x0 evaluates x0 once and inserts the result everywhere x appears:

Module introduces local variables to which values can be assigned:

With variables are read only:

With is faster than Module:

Block localizes values only; it does not substitute values. Module creates new symbols:

With allows substitution inside an unevaluated expression, preserving nested scopes:

Ordinary substitution does not preserve scoping:

Possible Issues  (1)

With is a scoping construct; variables are renamed in nested scopes:

Build the function from its elements to avoid the renaming:

Neat Examples  (2)

Find a zero of an arbitrary function using Newton's method:

Find a fixed point:

A version of With where the initializer is within the scope of the local variable:

Here the f inside the function definition is not inside its own scope:

Wolfram Research (1991), With, Wolfram Language function, https://reference.wolfram.com/language/ref/With.html.

Text

Wolfram Research (1991), With, Wolfram Language function, https://reference.wolfram.com/language/ref/With.html.

CMS

Wolfram Language. 1991. "With." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/With.html.

APA

Wolfram Language. (1991). With. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/With.html

BibTeX

@misc{reference.wolfram_2024_with, author="Wolfram Research", title="{With}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/With.html}", note=[Accessed: 13-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_with, organization={Wolfram Research}, title={With}, year={1991}, url={https://reference.wolfram.com/language/ref/With.html}, note=[Accessed: 13-December-2024 ]}