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Template:Math/tests

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alt = This template is under construction...
This template is under construction.            

Please do not use this unfinished and/or still unreliable template.            

This is a testing template for the {{math}} OEIS Wiki utility template. (This is a work in progress...!)

Tests

Polynomial

The code

{{indent}}{{math|''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1|&&}}, <!--
-->{{math|''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1|$$}}

yields the display style HTML+CSS, display style LaTeX

     
x 3x 2 + 1
,

while the code

: before {{math|''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1|&}}, <!--
-->{{math|''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1|$}} after

yields the text style HTML+CSS, text style LaTeX

before 
x 3  −  x 2  +  1
, after

Absolute value of polynomial

The code

{{indent}}{{math|{{abs| ''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1 |HTM}}|&&}}, <!--
-->{{math|{{abs| ''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1|TEX}}|$$}}

yields the display style HTML+CSS, display style LaTeX

     
x 3x 2 + 1
,

while the code

: before {{math|{{abs| ''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1 |HTM}}|&}}, <!--
-->{{math|{{abs| ''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1 |TEX}}|$}} after

yields the text style HTML+CSS, text style LaTeX

before 
x 3  −  x 2  +  1
, after

Absolute value of rational function

The code

{{indent}}{{math|{{abs| {{frac| 1 | ''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1 |HTM}} |HTM}}|&&}}, <!--
-->{{math|{{abs| {{frac| 1 | ''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1 |TEX}} |TEX}}|$$}}

yields the display style HTML+CSS, display style LaTeX

     
1
x 3x 2 + 1
,

while the code

: before {{math|{{abs| {{frac| 1 | ''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1 |htm}} |HTM}}|&}}, <!--
-->{{math|{{abs| {{frac| 1 | ''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1 |TEX}} |tex}}|$}} after

yields the text style HTML+CSS, text style LaTeX

before 
1
x 3  −  x 2  +  1
, after

Definition of derivative

The code

{{indent}}{{math|

  ''f{{sp|2}}{{sym|prime}}''{{sp|2}}(''x'') {{=}} ''d'' iff {{sym|forall}}{{sp|-4}}''{{Gr|epsilon}}'', {{sym|exists}}''{{Gr|delta}}'' s.t. 
  0 < {{abs|{{Gr|Delta}}{{thinsp}}''x''|htm}} < ''{{Gr|delta}}'' with 
  {{abs| 
    {{frac
    | {{thinsp}}''f''{{sp|3}}(''x'' {{op|+}} {{Gr|Delta}}{{thinsp}}''x'') {{op|-}} ''f''{{sp|3}}(''x'') 
    | {{Gr|Delta}}{{thinsp}}''x'' 
    |HTM}} {{op|-}} ''d'' 
  |HTM}} < ''{{Gr|epsilon}}''

|&&}}

yields the display style HTML+CSS

     
f  ′  (x) = d iff ∀ϵ, ∃δ s.t. 0 <
| Δ x |
< δ with
f (x + Δ x) − f (x)
Δ x
d
< ϵ

while the code

{{indent}}{{math| 

  ''f{{sym|prime|tex}}'' (''x'') {{=}} ''d'' \text{ iff } {{sym|forall|tex}}''{{Gr|epsilon|tex}}'',\, {{sym|exists|tex}}''{{Gr|delta|tex}}'' \text{ s.t. }  
  0 < {{abs| {{Gr|Delta|tex}}{{thinsp|tex}}''x'' |tex}} < ''{{Gr|delta|tex}}'' \text{ with } 
  {{abs|
    {{frac|''f''(''x'' {{op|+|tex}} {{Gr|Delta|tex}}{{thinsp|tex}}''x'') {{op|-|tex}} ''f''(''x'') 
    | {{Gr|Delta|tex}}{{thinsp|tex}}''x''
    |TEX}} {{op|-|tex}} ''d''
  |TEX}} < ''{{Gr|epsilon|tex}}''

|$$}}

yields the display style LaTeX

     

Fibonacci polynomials

The code

{{indent}}{{math|F{{sp|-2}}{{sub|''n''}}{{sp|1}}(''x'') {{sym|def}}
{{cases|begin}}
  {{&}}0,{{&}}if ''n'' {{=}} 0, {{\\}}
  {{&}}1,{{&}}if ''n'' {{=}} 1, {{\\}}
  {{&}}F{{sp|-2}}{{sub|''n''{{sp|1}}{{op|-}}1}}{{sp|1}}(''x'') {{op|+}} ''x''{{sp|3}}F{{sp|-2}}{{sub|''n''{{sp|1}}{{op|-}}2}}{{sp|1}}(''x''),{{sp|quad}}{{&}}if ''n'' {{rel|ge}} 2. 
{{cases|end}}
|tex = {\rm F}_{n}(x) := 
{{begin|cases|tex}}
  0, & \text{if } n = 0, \\
  1, & \text{if } n = 1, \\
  {\rm F}_{n-1}(x) + x \, {\rm F}_{n-2}(x), & \text{if } n \geq 2.
{{end|cases|tex}}
|&&}}

yields the display style HTML+CSS (with the && option):

     
Fn (x) :=
⎧  
⎨  
⎩  
0,if n = 0,
1,if n = 1,
Fn  − 1 (x) + x Fn  − 2 (x), if n ≥ 2.

or yields the display style LaTeX (with the $$ option):

     

Maxwell’s equations[1] (as differential equations)

The code

{{indent}}{{math|
{{align|begin}}
  {{op|curl}}{{vec|B|b}} {{op|-}} {{frac|1|''c''|HTM}}{{sp|2}}{{partial|{{vec|E|b}}|''t''|HTM}} {{&=}} {{frac|4{{Gr|pi}}|''c''|HTM}}{{sp|1}}{{vec|J|b}} {{\\}}
  {{op|divergence}}{{vec|E|b}} {{&=}} 4{{Gr|pi}}{{sp|3}}''{{Gr|rho}}'' {{\\}}
  {{op|curl}}{{vec|E|b}} {{op|+}} {{frac|1|''c''|HTM}}{{sp|2}}{{partial|{{vec|B|b}}|''t''|HTM}} {{&=}} {{vec|0|b}} {{\\}}
  {{op|divergence}}{{vec|B|b}} {{&=}} 0
{{align|end}}
|tex = 
{{align|begin|tex}}
  {{op|curl|tex}}{{vec|B|b|tex}} - {{frac|1|c|TEX}} {{partial|{{vec|E|b|tex}}|t|TEX}} {{&=|tex}} {{frac|4{{Gr|pi|tex}}|c|TEX}} {{vec|J|b|tex}} {{\\|tex}}
  {{op|divergence|tex}}{{vec|E|b|tex}} {{&=|tex}} 4{{Gr|pi|tex}} {{Gr|rho|tex}} {{\\|tex}}
  {{op|curl|tex}}{{vec|E|b|tex}} + {{frac|1|c|TEX}} {{partial|{{vec|B|b|tex}}|t|TEX}} {{&=|tex}} {{vec|0|b|tex}} {{\\|tex}}
  {{op|divergence|tex}}{{vec|B|b|tex}} {{&=|tex}} 0
{{align|end|tex}}
|&&}}

yields the display style HTML+CSS (with the && option):

     
∇ × B
1
c
  
E
t
 = 
c
J
∇ ∙ E  =  4π ρ
∇ × E +
1
c
  
B
t
 =  0
∇ ∙ B  =  0

or yields the display style LaTeX (with the $$ option):

     

Einstein's field equations[2]

The code

{{indent}}{{math|

<!-- 
-->''G''{{sub|''{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} {{op|+}} ''g''{{sub|''{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} {{Gr|Lambda}} {{=}} <!--
-->''R''{{sub|''{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} {{op|-}} {{frac|1|2|HTM}} ''g''{{sub|''{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} ''R'' <!--
-->{{op|+}} ''g''{{sub|''{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} {{Gr|Lambda}} {{=}} <!--
-->{{frac|8{{Gr|pi}}''G''|''c''{{^|4}}|HTM}}{{sp|1}}''T''{{sub|''{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}}

|&&}}

yields the display style HTML+CSS

     
Gμ ν + gμ ν Λ = Rμ ν
1
2
gμ ν R + gμ ν Λ =
G
c 4
Tμ ν

while the code

{{indent}}{{math|

G_{\mu \nu} + g_{\mu \nu} \Lambda {{=}} R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} \, R + g_{\mu \nu} \Lambda {{=}} \frac{8 \pi G}{c^4} T_{\mu \nu}

|$$}}

yields the display style LaTeX

     

Riemann curvature tensor

The code

{{indent}}{{math|<!-- LaTeX doesn't seem to italicize lambda, so we won't! --><!-- Cumbersome code... -->

<!-- 
-->''R''{{sup|''{{Gr|rho}}''}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} {{=}} <!-- 
-->{{sym|partial}}{{sub|''{{Gr|mu}}''}} {{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|nu}}''}} {{op|-}} <!--
-->{{sym|partial}}{{sub|''{{Gr|nu}}''}} {{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|mu}}''}} {{op|+}} <!--
-->{{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|{{Gr|lambda}}{{sp|1}}''{{Gr|mu}}''}} <!--
-->{{Gr|Gamma}}{{sup|{{Gr|lambda}}}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|nu}}''}} {{op|-}} <!--
-->{{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|{{Gr|lambda}}{{sp|1}}''{{Gr|nu}}''}} <!--
-->{{Gr|Gamma}}{{sup|{{Gr|lambda}}}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|mu}}''}}<!-- 
-->

|&&}}
yields the display style HTML+CSS (italicized lambda 
λ
doesn't look good, so we don't italicize it!)
     
Rρσ μ ν = ∂μ Γρσ ν − ∂ν Γρσ μ + Γρλ μ Γλσ ν − Γρλ ν Γλσ μ

while the code

{{indent}}{{math|
<!-- 
-->{ R^{\rho} }_{\sigma \mu \nu} {{=}} 
\partial_{\mu} { \Gamma^{\rho} }_{\sigma \nu} - 
\partial_{\nu} { \Gamma^{\rho} }_{\sigma \mu} + 
{ \Gamma^{\rho} }_{\lambda \mu} \, { \Gamma^{\lambda} }_{\sigma \nu} - 
{ \Gamma^{\rho} }_{\lambda \nu} \, { \Gamma^{\lambda} }_{\sigma \mu}<!-- 
-->
|$$}}

yields the display style LaTeX

     

Cardano formula[3]

The code

The reduced cubic equation {{math|''y''{{^|3}} {{op|+}} 3''py'' {{op|+}} 2''q'' {{=}} 0|&}} has one real and two complex solutions when <!--
-->{{math|''D'' {{=}} ''q''{{^|2}} {{op|+}} ''p''{{^|3}} {{rel|>}} 0|&}}. These are given by Cardan{{'}}s formula as{{nl}}

{{indent}}{{math|<!--
-->''y''{{sub|1}} {{=}} ''u'' {{op|+}} ''v'',{{sp|quad}}<!--
-->''y''{{sub|2}} {{=}} {{op|-}} {{frac|''u'' {{op|+}} ''v''|2|HTM}} {{op|+}} ''i''{{sp|3}}{{frac|{{sqrt|3}}|2|HTM}}{{sp|3}}(''u'' {{op|-}} ''v''),{{sp|quad}}<!--
-->''y''{{sub|3}} {{=}} {{op|-}} {{frac|''u'' {{op|+}} ''v''|2|HTM}} {{op|-}} ''i''{{sp|3}}{{frac|{{sqrt|3}}|2|HTM}}{{sp|3}}(''u'' {{op|-}} ''v''),
|&&}}

where

{{indent}}{{math|<!--
-->''u'' {{=}} {{root|{{op|-}} ''q'' {{op|+}} {{sqrt|''q''{{^|2}} {{op|+}} ''p''{{^|3}}}}|3|HTM}},{{sp|quad}}<!--
-->''v'' {{=}} {{root|{{op|-}} ''q'' {{op|-}} {{sqrt|''q''{{^|2}} {{op|+}} ''p''{{^|3}}}}|3|HTM}}. 
|&&}}

yields the HTML+CSS:

The reduced cubic equation 
y 3  +  3py  +  2q = 0
has one real and two complex solutions when 
D = q 2  +  p 3   >   0
. These are given by Cardan’s formula as
     
y1 = u + v, y2 = −
u + v
2
+ i
2  3
2
 (uv), y3 = −
u + v
2
i
2  3
2
 (uv),

where

     
u =
3  − q +
2  q 2 + p 3
, v =
3  − q
2  q 2 + p 3
.

The code

The reduced cubic equation {{math|''y''{{^|3|tex}}{{op|+}}3''py''{{op|+}}2''q'' {{=}} 0|$}} has one real and two complex solutions when <!--
-->{{math|''D'' {{=}} ''q''{{^|2|tex}}{{op|+}}''p''{{^|3|tex}} {{rel|>|tex}} 0|$}}. These are given by Cardan{{'}}s formula as{{nl}}
 
{{indent}}{{math|<!--

-->''y''{{sub|1|tex}} {{=}} ''u''{{op|+}}''v'',{{sp|quad|tex}}<!--
-->''y''{{sub|2|tex}} {{=}} {{op|-}}{{frac|''u''{{op|+}}''v''|2|TEX}}{{op|+}}''i''{{sp|3|tex}}{{frac|{{sqrt|3|tex}}|2|TEX}}{{sp|3|tex}}(''u'' {{op|-}} ''v''),{{sp|quad|tex}}<!--
-->''y''{{sub|3|tex}} {{=}} {{op|-}}{{frac|''u''{{op|+}}''v''|2|TEX}}{{op|-}}''i''{{sp|3|tex}}{{frac|{{sqrt|3|tex}}|2|TEX}}{{sp|3|tex}}(''u'' {{op|-}} ''v''),<!--
-->|$$}}

where

{{indent}}{{math|<!--
-->''u'' {{=}} {{root|{{op|-}}''q''{{op|+}}{{sqrt|''q''{{^|2|tex}}{{op|+}}''p''{{^|3|tex}}|tex}}|3|TEX}},{{sp|quad|tex}}<!--
-->''v'' {{=}} {{root|{{op|-}}''q''{{op|-}}{{sqrt|''q''{{^|2|tex}}{{op|+}}''p''{{^|3|tex}}|tex}}|3|TEX}}.<!--
-->  
|$$}}

yields the LaTeX:

The reduced cubic equation has one real and two complex solutions when. These are given by Cardan’s formula as

     

where

     

Curvature of plane curve

The code

The curvature at any point of the plane curve {{math|''C''|&}} given by {{math|''r''{{sp|1}}(''t'') {{=}} (''x''{{sp|1}}(''t''),{{sp|3}}''y''{{sp|1}}(''t''))|&}} is{{nl}}

{{indent}}{{math|

''{{Greek|kappa}}'' {{=}} {{frac
| ''x''{{op|1st}}{{sp|1}}''y''{{op|2nd}} {{op|-}} ''y''{{op|1st}}{{sp|1}}''x''{{op|2nd}}
| {{big|(}}({{sp|1}}''x''{{op|1st}}){{^|2}} {{op|+}} ({{sp|2}}''y''{{op|1st}}){{^|2}}{{sp|1}}{{big|)}}{{^|3/2}} 
|HTM}}{{sp|1}}.

|&&}}

yields the HTML+CSS:

The curvature at any point of the plane curve 
C
given by 
r (t) = (x (t), y (t))
is
     
κ =
xyyx
(( x) 2 + (  y) 2) 3/2
 .

The code

The curvature at any point of the plane curve {{math|''C''|$}} given by {{math|''r''(''t'') {{=}} (''x''(''t''),{{sp|3|tex}}''y''(''t''))|$}} is

{{indent}}{{math|

''{{Greek|kappa|tex}}'' {{=}} {{frac
| ''x''{{op|1st|tex}}''y''{{op|2nd|tex}} {{op|-|tex}} ''y''{{op|1st|tex}}''x''{{op|2nd|tex}}
| {{big|(|tex}}(''x''{{op|1st|tex}}){{^|2|tex}} {{op|+|tex}} (''y''{{op|1st|tex}}){{^|2|tex}}{{big|)|tex}}{{^|3/2|tex}} 
|TEX}}.

|$$}}

yields the LaTeX:

The curvature at any point of the plane curve given by is

     

Catalan numbers

The code

{{indent}}{{math|<!--

-->''C''{{sub|''n''}} {{=}} {{frac|1|''n'' + 1|HTM}} {{binom|2''n''|''n''|HTM}} {{=}} {{frac|(2''n'')!|(''n'' + 1)!{{sp|3}}''n''!|HTM}} {{=}} <!--
-->{{prod|''k''{{=}}2|''n''|{{frac|''n'' + ''k''|''k''|HTM}}|HTM}},{{sp|quad}}''n'' {{rel|ge}} 0.<!--

-->|&&}}

yields the display style HTML+CSS

     
Cn =
1
n + 1
(
2n
n
) =
(2n)!
(n + 1)! n!
=
n
k  = 2
  
n + k
k
, n ≥ 0.

The code

{{indent}}{{math|<!--

-->''C''{{sub|''n''|TEX}} {{=}} {{frac|1|''n'' + 1|TEX}} {{binom|2''n''|''n''|TEX}} {{=}} {{frac|(2''n'')!|(''n'' + 1)!{{sp|3|tex}}''n''!|TEX}} {{=}} <!--
-->{{prod|''k''{{=}}2|''n''|{{frac|''n'' + ''k''|''k''|TEX}}|TEX}},{{sp|quad|TEX}}''n'' {{rel|ge|TEX}} 0.<!--

-->|$$}}

yields the display style LaTeX

     

Summations and integrals

Mellin–Barnes integral[4]

Cf. Mellin–Barnes integralDLMF, NIST Project.

The code

{{indent}}{{math|
''B''{{sub|''n''}}{{sp|1}}(''x'') {{=}} {{frac|1|2{{Gr|pi}}{{sp|1}}''i''}}{{sp|6}}<!--
-->{{int|{{op|-}}''c''{{op|-}}''i''{{sp|1}}infty|{{op|-}}''c''{{op|+}}''i''{{sp|1}}infty
|(''x'' {{op|+}} ''t''){{^|''n''}} {{(|{{frac|{{Gr|pi}}|sin{{sp|1}}({{Gr|pi}}{{sp|1}}''t''{{sp|1}})}}|)|Big}}{{^|{{^|2|100%}}}} {{d|''t''}}|HTM}}.
|&&}}

yields the display style HTML+CSS

     
Bn (x) =
1
2π i
 − c + i
 − c − i
 (x + t)n
  
π
sin (π t )
  
 2 dt.

The code

{{indent}}{{math|
''B''{{sub|''n''|tex}}(''x'') {{=}} {{frac|1|2''{{Gr|pi|tex}}{{sp|1|tex}}i''|tex}} <!--
-->{{int|{{op|-}}''c''{{op|-}}''i''{{sp|1|tex}}infty|{{op|-}}''c''{{op|+}}''i''{{sp|1|tex}}infty
|(''x'' {{op|+}} ''t''){{^|''n''|tex}} {{(|{{frac|{{Gr|pi|tex}}|\sin{{sp|1|tex}}({{Gr|pi|tex}}{{sp|1|tex}}''t'')|tex}}|)|Big|tex}}{{^|2|tex}} <!--
-->{{d|''t''|tex}}|TEX}}.
|$$}}

yields the display style LaTeX

     

Elliptic integrals

Cf. Elliptic integralsDLMF, NIST Project.

The code

{{indent}}{{math|
''R''{{sub|{{sp|1}}{{op|-}}''a''}}{{sp|1}}({{vec|b|b}}; {{vec|z|b}}) {{=}}
 
{{frac
| 4{{sp|1}}{{Gr|Gamma}}(''b''{{sub|1}} {{op|+}} ''b''{{sub|2}} {{op|+}} ''b''{{sub|3}})
| {{Gr|Gamma}}(''b''{{sub|1}}){{sp|2}}{{Gr|Gamma}}(''b''{{sub|2}}){{sp|2}}{{Gr|Gamma}}(''b''{{sub|3}})
}}{{sp}} 
{{int|0|{{Gr|pi}}{{op|/}}2|HTM}}{{int|0|{{Gr|pi}}{{op|/}}2|HTM}}{{sp|6}}<!-- 
-->{{(|{{sum|''i''{{=}}1|3|''z''{{sub|''j''}}{{sp|2}}''l''{{subsup|{{sp|-2}}''j''|2}}|HTM}}|)|Bigg}}{{^|{{^|{{^|{{op|-}}''a''|100%}}|100%}}}}{{sp|6}}<!-- 
-->{{(|{{prod|''j''{{=}}1|3| ''l''{{subsup|''j''|2''b''{{sub|''j''}}{{sp|1}}{{op|-}}1}}|HTM}}|)|Bigg}}{{sp|6}}<!-- 
-->{{op|sin}}{{sp|1}}''{{Gr|theta}}''{{sp|3}}{{d|''{{Gr|theta}}''}}{{sp|3}}{{d|''{{Gr|phi}}''}},<!--
 
-->{{sp|quad}}''b''{{sub|''j''}} {{rel|>}} 0, {{sym|Re}}{{sp|1}}''z''{{sub|''j''}} {{rel|>}} 0.
|&&}}
yields the display style HTML+CSS (The exponent 
 − a
needs to be raised higher more conveniently!)

     
Error: String exceeds 10,000 character limit.

The code (broken in two parts to avoid the pesky: “Error: String exceeds 10,000 character limit.”)

{{indent}}{{math|
''R''{{sub|{{sp|1}}{{op|-}}''a''}}{{sp|1}}({{vec|b|b}}; {{vec|z|b}}) {{=}}
 
{{frac
| 4{{sp|1}}{{Gr|Gamma}}(''b''{{sub|1}} {{op|+}} ''b''{{sub|2}} {{op|+}} ''b''{{sub|3}})
| {{Gr|Gamma}}(''b''{{sub|1}}){{sp|2}}{{Gr|Gamma}}(''b''{{sub|2}}){{sp|2}}{{Gr|Gamma}}(''b''{{sub|3}})
}}|&&}}{{sp}}{{math| 
{{int|0|{{Gr|pi}}{{op|/}}2|HTM}}{{int|0|{{Gr|pi}}{{op|/}}2|HTM}}{{sp|6}}<!-- 
-->{{(|{{sum|''i''{{=}}1|3|''z''{{sub|''j''}}{{sp|2}}''l''{{subsup|{{sp|-2}}''j''|2}}|HTM}}|)|Bigg}}{{^|{{^|{{^|{{op|-}}''a''|100%}}|100%}}}}{{sp|6}}<!-- 
-->{{(|{{prod|''j''{{=}}1|3| ''l''{{subsup|''j''|2''b''{{sub|''j''}}{{sp|1}}{{op|-}}1}}|HTM}}|)|Bigg}}{{sp|6}}<!-- 
-->{{op|sin}}{{sp|1}}''{{Gr|theta}}''{{sp|3}}{{d|''{{Gr|theta}}''}}{{sp|3}}{{d|''{{Gr|phi}}''}},<!--
 
-->{{sp|quad}}''b''{{sub|''j''}} {{rel|>}} 0, {{sym|Re}}{{sp|1}}''z''{{sub|''j''}} {{rel|>}} 0.
|&&}}
yields the display style HTML+CSS (The exponent 
 − a
needs to be raised higher more conveniently!)

     
R  − a (b; z) =
4 Γ(b1 + b2 + b3)
Γ(b1)  Γ(b2)  Γ(b3)
π / 2
0
π / 2
0
3
i  = 1
  zj  l2j
  − a
3
j  = 1
   l2bj −1j
 sin θdθdϕ, bj > 0, ℜ zj > 0.

The code

{{indent}}{{math|
''R''{{sub|{{op|-}}''a''|tex}}{{sp|1|tex}}({{vec|b|b|tex}}; {{vec|z|b|tex}}) {{=}} <!--
-->{{frac
|4{{sp|1|tex}}{{Gr|Gamma|tex}}(''b''{{sub|1|tex}}{{op|+}}''b''{{sub|2|tex}}{{op|+}}''b''{{sub|3|tex}})
|{{Gr|Gamma|tex}}(''b''{{sub|1|tex}}){{sp|2|tex}}{{Gr|Gamma|tex}}(''b''{{sub|2|tex}}){{sp|2|tex}}{{Gr|Gamma|tex}}(''b''{{sub|3|tex}})
|tex}}<!--
-->{{sp|3|tex}}{{int|0|{{Gr|pi|tex}}{{op|/|tex}}2|TEX}}{{int|0|{{Gr|pi|tex}}{{op|/|tex}}2|TEX}}<!--
--> {{(|{{sum|''i''{{=}}1|3| ''z''{{sub|''j''|tex}}{{sp|1|tex}}''l''{{subsup|''j''|2|tex}}|TEX}}|)|Bigg|tex}}{{^|{{op|-}}''a''|tex}} <!--
--> {{(|{{prod|''j''{{=}}1|3| ''l''{{subsup|''j''|2''b''{{sub|''j''|tex}}{{op|-}}1|tex}}|TEX}}|)|Bigg|tex}}<!--
-->{{sp|3|tex}}{{op|sin|tex}}''{{Gr|theta|tex}}''{{sp|3|tex}}{{d|''{{Gr|theta|tex}}''|tex}}{{sp|3|tex}}{{d|''{{Gr|phi|tex}}''|tex}},<!--
-->{{sp|quad|tex}}''b''{{sub|''j''|tex}} {{rel|>|tex}} 0, {{sym|Re|tex}}{{sp|1|tex}}''z''{{sub|''j''|tex}} {{rel|>|tex}} 0.
|$$}}

yields the display style LaTeX

     

γ/π

The code (see A301813 for decimal expansion of
γ/π
)
{{indent}}{{math|{{frac|{{Gr|gamma}}|{{Gr|pi}}|HTM}} {{=}} {{int|{{op|-}}infty|{{op|+}}infty|HTM}} {{op|-}} log {{root|''z''{{^|2}} {{op|+}} {{tfrac|1|4}}|4|HTM}} {{op|sech}}{{sp|1}}({{Gr|pi}}{{sp|1}}''z''){{^|2}} {{d|''z''}}
|tex = \frac{\gamma}{\pi} = \int_{-\infty}^{+\infty} - \log \sqrt[4]{ z^2 + \tfrac{1}{4} } {{op|sech|tex}}(\pi z)^2 dz 
|$$}}

yields the display style HTML+CSS (with the && option)

     
γ
π
=
 + 
 − 
− log
4  z 2 +
1
4
sech (π z) 2 dz

or yields the display style LaTeX (with the $$ option):

     

Delta function[5]

The code

{{indent}}{{math|''{{Gr|delta}}''{{sp|1}}(''x'') {{=}} <!--
-->{{lim|''n''->infty|HTM}} {{frac|1|2{{Gr|pi}}|HTM}} {{frac|{{fn|sin}} {{(|{{(|''n'' + {{tfrac|1|2}}|)|big}} ''x''|)|big [ ]}}|{{fn|sin}} {{(|{{tfrac|1|2}} ''x''|)|big}}|HTM}}.
| tex = \delta(x) = \lim_{n \to \infty} \frac{1}{2\pi} \frac{\sin [(n + \frac{1}{2}) x]}{\sin (\frac{1}{2} x)}. 
|&&}}

yields the display style HTML+CSS

     
δ (x) =
1
sin
  n +
1
2
  
x
sin
  
1
2
x  
.

The code

{{indent}}{{math|''{{Gr|delta|tex}}''(''x'') {{=}} {{lim|''n''->infty|TEX}} {{frac|1|2{{Gr|pi|tex}}|tex}} {{frac|{{fn|sin|tex}} [(''n'' + {{tfrac|1|2|tex}}) ''x'']|{{fn|sin|tex}} ({{tfrac|1|2|tex}} ''x'')|tex}}.|$$}}

yields the display style LaTeX

     

The code

: <math>\delta(x) = \lim_{n \to \infty} \frac{1}{2\pi} \frac{\sin [(n + \frac{1}{2}) x]}{\sin (\frac{1}{2} x)}.</math>

yields the display style LaTeX

BPP formula

The code

: {{math
|{{Gr|pi}} {{=}} {{sum|''k''{{=}}0|infty|HTM}} {{sqbrack|Big l}} {{frac|1|16{{^|''k''}}|HTM}} <!--
-->{{paren|Big l}}{{frac|4|8''k'' + 1|HTM}} {{op|-}} {{frac|2|8''k'' + 4|HTM}} {{op|-}} {{frac|1|8''k'' + 5|HTM}} {{op|-}} {{frac|1|8''k'' + 6|HTM}}{{paren|Big r}} {{sqbrack|Big r}},
|tex = \pi = \sum_{k{{=}}0}^{\infty} \left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right], 
|&&}}

yields the display style HTML+CSS (the {{(|...|)}} template needs more work!)

π =
k  = 0
  
1
16k
4
8k + 1
2
8k + 4
1
8k + 5
1
8k + 6
,

and with the $$ option yields the display style LaTeX

Notes