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Through[p[f,g,…][x,y,…]]

gives p[f[x,y,…],g[x,y,…],…].

Through[expr,h]

performs the transformation wherever h occurs in the head of expr.

Details
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Basic Examples  
Scope  
Applications  
Properties & Relations  
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Through

Through[p[f,g,…][x,y,…]]

gives p[f[x,y,…],g[x,y,…],…].

Through[expr,h]

performs the transformation wherever h occurs in the head of expr.

Details

  • Through distributes operators that appear inside the heads of expressions.
  • Parallelize[Through[p[f,g,…][x,y,…]]] computes Through[p[f,g,…][x,y,…]] in parallel on all subkernels. »

Examples

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Basic Examples  (2)

Apply all functions in a list to the same argument:

Apply a sum of functions by forming the sum of the individual applications:

Scope  (2)

By default, the operation is performed for any function in the head:

Perform the operation only if the head is a sum of terms:

Applications  (2)

Apply a differential operator:

Properties & Relations  (5)

Tabulate square roots of values without using auxiliary variables:

Compute Through in parallel:

Through[p[f,g,…][x]] is equivalent to Comap[p[f,g,…],x]:

Compare this to the operator form of Comap:

Through[p[f,g,…][x,y,…]] is equivalent to ComapApply[p[f,g,…],{x,y,…}]:

Compare this to the operator form of ComapApply:

Wolfram Research (1988), Through, Wolfram Language function, https://reference.wolfram.com/language/ref/Through.html.

Text

Wolfram Research (1988), Through, Wolfram Language function, https://reference.wolfram.com/language/ref/Through.html.

CMS

Wolfram Language. 1988. "Through." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Through.html.

APA

Wolfram Language. (1988). Through. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Through.html

BibTeX

@misc{reference.wolfram_2025_through, author="Wolfram Research", title="{Through}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Through.html}", note=[Accessed: 08-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_through, organization={Wolfram Research}, title={Through}, year={1988}, url={https://reference.wolfram.com/language/ref/Through.html}, note=[Accessed: 08-August-2025]}