Template:Math/tests

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 $L a T e X$

x 3 − x 2 + 1

,

{\begin{array}{l}\displaystyle {x^{3}-x^{2}+1}\end{array}}

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 $L a T e X$

before 

x 3 − x 2 + 1

,

\textstyle {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 $L a T e X$

x 3 − x 2 + 1

,

{\begin{array}{l}\displaystyle {\left\vert x^{3}-x^{2}+1\right\vert }\end{array}}

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 $L a T e X$

before 

x 3 − x 2 + 1

,

\textstyle {\left\vert x^{3}{\,}-{\,}x^{2}+1\right\vert }

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 $L a T e X$

1

x 3 − x 2 + 1

,

{\begin{array}{l}\displaystyle {\left\vert {\frac {1}{x^{3}-x^{2}+1}}\right\vert }\end{array}}

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 $L a T e X$

before 

1

x 3 − x 2 + 1

,

\textstyle {\left\vert {\frac {1}{x^{3}{\,}-{\,}x^{2}+1}}\right\vert }

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 $L a T e X$

${\begin{array}{l}\displaystyle {f^{\prime }(x)=d{\text{ iff }}{\forall }\epsilon ,\,{\exists }\delta {\text{ s.t. }}0<\vert \Delta \,x\vert <\delta {\text{ with }}\left\vert {\frac {f(x+\Delta \,x)-f(x)}{\Delta \,x}}-d\right\vert <\epsilon }\end{array}}$

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 $L a T e X$ (with the $$ option):

${\begin{array}{l}\displaystyle {{\rm {F}}_{n}(x):={\begin{cases}0,&{\text{if }}n=0,\\1,&{\text{if }}n=1,\\{\rm {F}}_{n-1}(x)+x\,{\rm {F}}_{n-2}(x),&{\text{if }}n\geq 2.\end{cases}}}\end{array}}$

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

=

4π

c

J

∇ ∙ E

= 4π ρ

∇ × E +

1

c

∂B

∂t

= 0

∇ ∙ B

= 0

or yields the display style $L a T e X$ (with the $$ option):

${\begin{array}{l}\displaystyle {\begin{aligned}{\nabla }\,{\times }\,\mathbf {B} -{\frac {1}{c}}{\dfrac {{\partial }^{}\mathbf {E} }{{\partial }{t}^{}}}&={\frac {4\pi }{c}}\mathbf {J} \\{\nabla }\,\mathbf {\cdot } \,\mathbf {E} &=4\pi \rho \\{\nabla }\,{\times }\,\mathbf {E} +{\frac {1}{c}}{\dfrac {{\partial }^{}\mathbf {B} }{{\partial }{t}^{}}}&=\mathbf {0} \\{\nabla }\,\mathbf {\cdot } \,\mathbf {B} &=0\end{aligned}}\end{array}}$

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 =

8πG

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 $L a T e X$

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

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 $L a T e X$

${\begin{array}{l}\displaystyle {{R^{\rho }}_{\sigma \mu \nu }=\partial _{\mu }{\Gamma ^{\rho }}_{\sigma \nu }-\partial _{\nu }{\Gamma ^{\rho }}_{\sigma \mu }+{\Gamma ^{\rho }}_{\lambda \mu }\,{\Gamma ^{\lambda }}_{\sigma \nu }-{\Gamma ^{\rho }}_{\lambda \nu }\,{\Gamma ^{\lambda }}_{\sigma \mu }}\end{array}}$

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

(u − v), y3 = −

u + v

2

− i

2√ 3

2

(u − v),

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 $L a T e X$ :

The reduced cubic equation $\textstyle {y^{3}+3py+2q=0}$ has one real and two complex solutions when $\textstyle {D=q^{2}+p^{3}{>}0}$ . These are given by Cardan’s formula as

${\begin{array}{l}\displaystyle {y_{1}{\!\,\!}=u+v,{\quad }y_{2}{\!\,\!}=-{\frac {u+v}{2}}+i{\,}{\frac {\sqrt {3}}{2}}{\,}(u-v),{\quad }y_{3}{\!\,\!}=-{\frac {u+v}{2}}-i{\,}{\frac {\sqrt {3}}{2}}{\,}(u-v),}\end{array}}$

where

${\begin{array}{l}\displaystyle {u={\sqrt[{3}]{-q+{\sqrt {q^{2}+p^{3}}}}},{\quad }v={\sqrt[{3}]{-q-{\sqrt {q^{2}+p^{3}}}}}.}\end{array}}$

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 =

x y − y x

(( 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 $L a T e X$ :

The curvature at any point of the plane curve $\textstyle {C}$ given by $\textstyle {r(t)=(x(t),{\,}y(t))}$ is

${\begin{array}{l}\displaystyle {\kappa ={\frac {xy-yx}{{\big (}(x)^{2}+(y)^{2}{\big )}^{3/2}}}.}\end{array}}$

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 $L a T e X$

${\begin{array}{l}\displaystyle {C_{n}{\!\,\!}={\frac {1}{n+1}}{\binom {2n}{n}}\!={\frac {(2n)!}{(n+1)!{\,}n!}}=\prod _{k=2}^{n}{\frac {n+k}{k}},{\quad }n\geq 0.}\end{array}}$

Summations and integrals

Mellin–Barnes integral^[4]

Cf. Mellin–Barnes integral—DLMF, 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 d t.

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 $L a T e X$

${\begin{array}{l}\displaystyle {B_{n}{\!\,\!}(x)={\frac {1}{2\pi {\;\;\!\!\!}i}}\int _{-c-i{\;\;\!\!\!}\infty }^{-c+i{\;\;\!\!\!}\infty }\;{(x+t)^{n}{\Big (}{\frac {\pi }{\sin {\;\;\!\!\!}(\pi {\;\;\!\!\!}t)}}{\Big )}^{2}d^{}t}.}\end{array}}$

Elliptic integrals

Cf. Elliptic integrals—DLMF, 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 l 2j

⎞

⎟

⎠

− a

⎛

⎜

⎝

3

∏

j = 1

  l 2bj −1j

⎞

⎟

⎠

sin θ d θ d ϕ, 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 $L a T e X$

${\begin{array}{l}\displaystyle {R_{-a}{\!\,\!}{\;\;\!\!\!}(\mathbf {b} ;\mathbf {z} )={\frac {4{\;\;\!\!\!}\Gamma (b_{1}{\!\,\!}+b_{2}{\!\,\!}+b_{3}{\!\,\!})}{\Gamma (b_{1}{\!\,\!}){\;\!}\Gamma (b_{2}{\!\,\!}){\;\!}\Gamma (b_{3}{\!\,\!})}}{\,}\int _{0}^{\pi /2}\;\int _{0}^{\pi /2}\;{\Bigg (}\sum _{i=1}^{3}z_{j}{\!\,\!}{\;\;\!\!\!}l_{j}^{2}{\Bigg )}^{-a}{\Bigg (}\prod _{j=1}^{3}l_{j}^{2b_{j}{\!\,\!}-1}{\Bigg )}{\,}{\sin }\,\theta {\,}d^{}\theta {\,}d^{}\phi ,{\quad }b_{j}{\!\,\!}{>}0,{\Re }{\;\;\!\!\!}z_{j}{\!\,\!}{>}0.}\end{array}}$

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 d z

or yields the display style $L a T e X$ (with the $$ option):

${\begin{array}{l}\displaystyle {{\frac {\gamma }{\pi }}=\int _{-\infty }^{+\infty }-\log {\sqrt[{4}]{z^{2}+{\tfrac {1}{4}}}}\operatorname {sech} \,(\pi z)^{2}dz}\end{array}}$

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

2π

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 $L a T e X$

${\begin{array}{l}\displaystyle {\delta (x)=\lim _{n{\to }\infty }{\frac {1}{2\pi }}{\frac {{\sin }\,[(n+{\tfrac {1}{2}})x]}{{\sin }\,({\tfrac {1}{2}}x)}}.}\end{array}}$

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 $L a T e X$

\delta (x)=\lim _{n\to \infty }{\frac {1}{2\pi }}{\frac {\sin[(n+{\frac {1}{2}})x]}{\sin({\frac {1}{2}}x)}}.

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

16 k

⎛

⎝

4

8k + 1

−

2

8k + 4

−

1

8k + 5

−

1

8k + 6

⎞

⎠

⎤

⎦

,

and with the $$ option yields the display style $L a T e X$

{\begin{array}{l}\displaystyle {\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],}\end{array}}

Notes

↑ Maxwell’s equations#Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)—Wikipedia.org.
↑ Einstein's field equations—Wikipedia.org.
↑ Weisstein, Eric W., Cubic Formula, from MathWorld—A Wolfram Web Resource.
↑ Weisstein, Eric W., Mellin-Barnes Integral, from MathWorld—A Wolfram Web Resource.
↑ Weisstein, Eric W., Delta Function, from MathWorld—A Wolfram Web Resource.

[1] Maxwell’s equations#Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)—Wikipedia.org.

[2] Einstein's field equations—Wikipedia.org.

[3] Weisstein, Eric W., Cubic Formula, from MathWorld—A Wolfram Web Resource.

[4] Weisstein, Eric W., Mellin-Barnes Integral, from MathWorld—A Wolfram Web Resource.

[5] Weisstein, Eric W., Delta Function, from MathWorld—A Wolfram Web Resource.

[1]

[2]

[3]

[4]

[5]

Template:Math/tests

Contents

Tests

Polynomial

Absolute value of polynomial

Absolute value of rational function

Definition of derivative

Fibonacci polynomials

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

Einstein's field equations^[2]

Riemann curvature tensor

Cardano formula^[3]

Curvature of plane curve

Catalan numbers

Summations and integrals

Mellin–Barnes integral^[4]

Elliptic integrals

γ/π

Delta function^[5]

BPP formula

Notes

Navigation menu

Views

Personal tools

Navigation

Search

Tools

Template:Math/tests

Contents

Tests

Polynomial

Absolute value of polynomial

Absolute value of rational function

Definition of derivative

Fibonacci polynomials

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

Einstein's field equations[2]

Riemann curvature tensor

Cardano formula[3]

Curvature of plane curve

Catalan numbers

Summations and integrals

Mellin–Barnes integral[4]

Elliptic integrals

γ/π

Delta function[5]

BPP formula

Notes

Navigation menu

Search

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

Einstein's field equations^[2]

Cardano formula^[3]

Mellin–Barnes integral^[4]

Delta function^[5]