Template:Prod

The following documentation is located at Template:Prod/doc. [<Edit> Template:Prod/doc]
[⧼Purge⧽ Template:Prod]

This template have been optimized (in the case of HTML+CSS typesetting) for Firefox,
though with some compromises to obtain [more or less] acceptable results with MS Internet Explorer (or now MS Edge) and Google Chrome.

The {{prod}} (product operator) mathematical formatting template is used to obtain an HTML+CSS or a LaTeX formatted product expression.

Usage

{{prod|from|to}} (followed by multiplicand, as with LaTeX; format defaults to htm)

or

{{prod|from|to|format}} (followed by multiplicand, as with LaTeX)

or

{{prod|from|to|multiplicand}}

or

{{prod|from|to|multiplicand|format}}

where format is one of the following:

• htm: HTML+CSS markup (text style, with postscript limits) for product expression (default);
• HTM: HTML+CSS markup (display style, with underscript/overscript limits) for product expression;
• tex: LaTeX markup (text style, with postscript limits) for product expression;
• TEX: LaTeX markup (display style, with underscript/overscript limits) for product expression.

Examples

The code

: line of text above{{nl}}another line of text above{{nl}}before {{math|{{prod|''i''{{=}}1|''n''|''i''{{^|3}}}}|&}} after{{nl}}line of text below{{nl}}another line of text below

yields the text style HTML+CSS

line of text above
another line of text above
before 

∏ n i  = 1  i 3

after
line of text below
another line of text below

The code

: line of text above{{nl}}another line of text above{{nl}}before {{math|{{prod|''i''{{=}}1|''n''|''i''{{^|3|tex}}|tex}}|$}} after{{nl}}line of text below{{nl}}another line of text below

yields the text style LaTeX

line of text above
another line of text above
before$\textstyle {\prod _{i=1}^{n}i^{3}}$ after
line of text below
another line of text below

The code

{{math|{{prod|''i''{{=}}1|''n''|''i''{{^|3|HTM}}|HTM}}|&&}}

yields the display style HTML+CSS

n ∏   i  = 1   i 3

The code

{{math|{{prod|''i''{{=}}1|''n''|''i''{{^|3|TEX}}|TEX}}|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {\prod _{i=1}^{n}i^{3}}\end{array}}$

The code

before {{math|{{prod|''p''{{nl}}''p'' prime|<!-- upper -->|(1 {{op|-}} ''p''{{^|{{op|-}}{{sp|1}}''s''}}{{sp|1}})}}|&}} after

yields the text style HTML+CSS

before 

∏ p p prime p p prime  (1  −  p  −  s )

after

The code

before {{math|{{prod|\stackrel{p}{ p {\rm ~prime} }|<!-- upper -->|(1 \,-\, p^{- s})|tex}}|$}} after

yields the text style LaTeX

before$\textstyle {\prod _{\stackrel {p}{p{\rm {~prime}}}}^{}(1\,-\,p^{-s})}$ after

The code

{{math|{{prod|''p''{{nl}}''p'' prime|<!-- upper -->|(1 {{op|-}} ''p''{{^|{{op|-}}{{sp|1}}''s''}}{{sp|1}})|HTM}}|&&}}

yields the display style HTML+CSS

p p prime ∏   p p prime   (1 − p  −  s )

The code

{{math|{{prod|\stackrel{p}{ p {\rm ~prime} }|<!-- upper -->|(1 - p^{- s})|TEX}}|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {\prod _{\stackrel {p}{p{\rm {~prime}}}}^{}(1-p^{-s})}\end{array}}$

The code

: The factorial of {{math|''n''|tex = n|&}}, defined as <!--
-->{{math|''n''! {{sym|def}} {{prod|''i''{{=}}1|''n''|''i''}}
|tex = n! := \prod_{i=1}^{n} i
|&}} gives the number of permutations of {{math|''n''|tex = n|&}}.

yields the text style HTML+CSS

The factorial of

n

, defined as

n! := ∏ n i  = 1  i

gives the number of permutations of

n

.

and, with the $ argument, yields the text style LaTeX

The factorial of $\textstyle {n}$, defined as $\textstyle {n!:=\prod _{i=1}^{n}i}$ gives the number of permutations of $\textstyle {n}$.

Examples with fraction

The code

: before {{math|<!--
-->{{prod|''p''{{nl}}''p'' prime|<!-- upper --><!--
-->|{{frac|1|1 {{op|-}} ''p''{{^|{{op|-}}{{sp|1}}''s''}}|htm}}<!--
-->}}|&}} after

yields the text style HTML+CSS

before 

∏ p p prime p p prime   1 1  −  p  −  s

after

The code

: before {{math|{{prod|\stackrel{p}{ p {\rm ~prime} }|<!-- upper -->|{{frac|1|1 - ''p''{{^|- ''s''|tex}}|tex}}|tex}}|$}} after

yields the text style LaTeX

before$\textstyle {\prod _{\stackrel {p}{p{\rm {~prime}}}}^{}{\frac {1}{1-p^{-s}}}}$ after

The code

: {{math|<!--
-->{{prod|''p''{{nl}}''p'' prime|{{nl|2}}<!-- upper -->
   |{{frac|1|1 {{op|-}} ''p''{{^|{{op|-}}{{sp|1}}''s''}}|HTM}}<!--
-->|HTM}} |&&}}

yields the display style HTML+CSS

∏   p p prime    1 1 − p  −  s

The code

: {{math|{{prod|\stackrel{p}{ p {\rm ~prime} }|<!-- upper -->|\frac{1}{ 1 - p^{- s} }|TEX}}|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {\prod _{\stackrel {p}{p{\rm {~prime}}}}^{}{\frac {1}{1-p^{-s}}}}\end{array}}$

Examples with equality

The code

{{indent}}before {{math| ''{{Gr|zeta}}''{{sp|1}}(''s'') {{sym|def}} <!--

-->{{sum|''n''{{=}}1|infty
   | {{frac|1|''n''{{^|''s''}}|htm}} 
   }}<!--

--> {{=}} <!--

-->{{prod|''p''{{nl}}''p'' prime|<!-- upper -->
   | {{frac|1|1 {{op|-}} ''p''{{^|{{op|-}}''s''}}|htm}} 
   }},{{sp|quad}}{{sym|real}}(''s'') > 1,<!--

-->
|&}} after

yields the text style HTML+CSS

     before 

ζ (s) := ∑ ∞ n  = 1   1 n s = ∏ p p prime p p prime   1 1  −  p  − s , ℜ(s) > 1,

after

The code

before {{math|''{{Gr|zeta|tex}}''(''s'') \,{{sym|def|tex}}\, <!--

-->{{sum|''n''{{=}}1|infty
   | {{frac|1|''n''{{^|''s''|TEX}}|TEX}}<!--
-->|TEX}} {{=}} <!--
-->{{prod|\stackrel{p}{ p {\rm ~prime} }|<!-- upper -->|{{frac|1|1 - ''p''{{^|- ''s''|tex}}|tex}}|tex}},{{sp|quad|tex}}{{sym|real|tex}}(''s'') > 1,<!--

-->
|$}} after

yields the text style LaTeX

before$\textstyle {\zeta (s)\,:=\,\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{\stackrel {p}{p{\rm {~prime}}}}^{}{\frac {1}{1-p^{-s}}},{\quad }{\Re }(s)>1,}$ after

The code

{{indent}}{{math|''{{Gr|zeta}}''{{sp|1}}(''s'') {{sym|def}} <!--
-->{{sum|''n''{{=}}1|infty
   | {{frac|1|''n''{{^|''s''}}|HTM}}
   |HTM}}<!--
--> {{=}} <!--

-->{{prod|''p''{{nl}}''p'' prime|{{nl|2}}<!-- upper -->
   | {{frac|1|1 {{op|-}} ''p''{{^|{{op|-}}''s''}}|HTM}} 
   |HTM}},{{sp|quad}}{{sym|real}}(''s'') > 1.
|&&}}{{nl}}

{{indent}}{{math|''{{Gr|zeta}}''{{sp|1}}(''s'') {{sym|def}} <!--
-->{{sum|''n''{{=}}1|infty
   | {{frac|1|''n''{{^|''s''}}|HTM}}
   |HTM}}<!--
--> {{=}} <!--

-->{{prod|''p'' : ''p'' prime|<!-- upper -->
   | {{frac|1|1 {{op|-}} ''p''{{^|{{op|-}}''s''}}|HTM}} 
   |HTM}},{{sp|quad}}{{sym|real}}(''s'') > 1.
|&&}}

yields the display style HTML+CSS

     

ζ (s) :=   ∞ ∑   n  = 1    1 n s =   ∏   p p prime    1 1 − p  − s , ℜ(s) > 1.

     

ζ (s) :=   ∞ ∑   n  = 1    1 n s =   p : p prime ∏   p : p prime    1 1 − p  − s , ℜ(s) > 1.

while the code

{{math|''{{Gr|zeta|tex}}''(''s'') {{sym|def|tex}} <!--

-->{{sum|''n''{{=}}1|infty
   | {{frac|1|''n''{{^|''s''|TEX}}|TEX}}<!--
-->|TEX}} {{=}} <!--
-->{{prod|\stackrel{p}{ p {\rm ~prime} }|<!-- upper -->|{{frac|1|1 - ''p''{{^|- ''s''|TEX}}|TEX}}|TEX}},{{sp|quad|TEX}}{{sym|real|TEX}}(''s'') > 1.<!--

-->
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {\zeta (s):=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{\stackrel {p}{p{\rm {~prime}}}}^{}{\frac {1}{1-p^{-s}}},{\quad }{\Re }(s)>1.}\end{array}}$

Examples with nested products

The code

before {{math|<!--

-->{{sum|''i''{{=}}1|''p''<!--
-->|{{prod|''j''{{=}}1|''q''<!--
  -->|{{prod|''k''{{=}}1|''r''<!--
    -->|''a''{{sub|''i''{{sp|2}}''j''}}{{sp|2}}''b''{{sub|''j''{{sp|1}}''k''}}{{sp|3}}''c''{{sub|''k''{{sp|1}}''i''}}<!--
    -->|htm}}<!--
  -->|htm}}<!--
-->|htm}}<!--
--> 
|&}} after

yields the text style HTML+CSS

before 

∑ p i  = 1   ∏ q j  = 1   ∏ r k  = 1  ai  j  bj k ck i

after

while the code

before {{math|<!--

-->{{sum|''i''{{=}}1|''p''<!--
-->|{{prod|''j''{{=}}1|''q''<!--
  -->|{{prod|''k''{{=}}1|''r''<!--
    -->|''a''{{sub|''ij''|tex}}{{sp|2|tex}}''b''{{sub|''jk''|tex}}{{sp|3|tex}}''c''{{sub|''ki''|tex}}<!--
    -->|tex}}<!--
  -->|tex}}<!--
-->|tex}}<!--
--> 
|$}} after

yields the text style LaTeX

before$\textstyle {\sum _{i=1}^{p}\prod _{j=1}^{q}\prod _{k=1}^{r}a_{ij}{\!\,\!}{\;\!}b_{jk}{\!\,\!}{\,}c_{ki}{\!\,\!}}$ after

The code

{{math|<!--

-->{{sum|''i''{{=}}1|''p''<!--
-->|{{prod|''j''{{=}}1|''q''<!--
  -->|{{prod|''k''{{=}}1|''r''<!--
    -->|''a''{{sub|''i''{{sp|2}}''j''}}{{sp|2}}''b''{{sub|''j''{{sp|1}}''k''}}{{sp|3}}''c''{{sub|''k''{{sp|1}}''i''}}<!--
    -->|HTM}}<!--
  -->|HTM}}<!--
-->|HTM}}<!--
-->
|&&}}

yields the display style HTML+CSS

p ∑   i  = 1      q ∏   j  = 1      r ∏   k  = 1   ai  j  bj k ck i

while the code

{{math|<!--

-->{{sum|''i''{{=}}1|''p''<!--
-->|{{prod|''j''{{=}}1|''q''<!--
  -->|{{prod|''k''{{=}}1|''r''<!--
    -->|''a''{{sub|''ij''|TEX}}{{sp|2|TEX}}''b''{{sub|''jk''|TEX}}{{sp|3|TEX}}''c''{{sub|''ki''|TEX}}<!--
    -->|TEX}}<!--
  -->|TEX}}<!--
-->|TEX}}<!--
--> 
|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {\sum _{i=1}^{p}\prod _{j=1}^{q}\prod _{k=1}^{r}a_{ij}{\!\,\!}{\;\!}b_{jk}{\!\,\!}{\,}c_{ki}{\!\,\!}}\end{array}}$

See also

• {{sum}} (summation)
• {{prod}} (product)
• {{int}} (simple integral: {{integral}} with int as type argument)
• {{integral}} (type argument: int, iint, iiint, iiiint, idotsint, oint, varointclockwise, ointctrclockwise, oiint, oiiint)