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Template:Math/tests
This is a testing template for the {{math}} OEIS Wiki utility template. (This is a work in progress...!)
Contents
- 1 Tests
- 1.1 Polynomial
- 1.2 Absolute value of polynomial
- 1.3 Absolute value of rational function
- 1.4 Definition of derivative
- 1.5 Fibonacci polynomials
- 1.6 Maxwell’s equations[1] (as differential equations)
- 1.7 Einstein's field equations[2]
- 1.8 Riemann curvature tensor
- 1.9 Cardano formula[3]
- 1.10 Curvature of plane curve
- 1.11 Catalan numbers
- 1.12 Summations and integrals
- 2 γ/π
- 3 Notes
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|$$}}
yields the display style HTML+CSS, display style LaTeX
x 3 − x 2 + 1 |
while the code
: before {{math|''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1|&}}, <!-- -->{{math|''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1|$}} after
yields the text style HTML+CSS, text style LaTeX
- before
, afterx 3 − x 2 + 1
Absolute value of polynomial
The code
{{indent}}{{math|{{abs| ''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1 |HTM}}|&&}}, <!-- -->{{math|{{abs| ''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1|TEX}}|$$}}
yields the display style HTML+CSS, display style LaTeX
|
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
, afterx 3 − x 2 + 1
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}}|$$}}
yields the display style HTML+CSS, display style LaTeX
|
while the code
: before {{math|{{abs| {{frac| 1 | ''x''{{^|3}} {{op|-}} ''x''{{^|2}} {{op|+}} 1 |htm}} |HTM}}|&}}, <!-- -->{{math|{{abs| {{frac| 1 | ''x''{{^|3|tex}} {{op|-|tex}} ''x''{{^|2|tex}} {{op|+|tex}} 1 |TEX}} |tex}}|$}} after
yields the text style HTML+CSS, text style LaTeX
- before
, after1 x 3 − x 2 + 1
Definition of derivative
The code
{{indent}}{{math| ''f{{sp|2}}{{sym|prime}}''{{sp|2}}(''x'') {{=}} ''d'' iff {{sym|forall}}{{sp|-4}}''{{Gr|epsilon}}'', {{sym|exists}}''{{Gr|delta}}'' s.t. 0 < {{abs|{{Gr|Delta}}{{thinsp}}''x''|htm}} < ''{{Gr|delta}}'' with {{abs| {{frac | {{thinsp}}''f''{{sp|3}}(''x'' {{op|+}} {{Gr|Delta}}{{thinsp}}''x'') {{op|-}} ''f''{{sp|3}}(''x'') | {{Gr|Delta}}{{thinsp}}''x'' |HTM}} {{op|-}} ''d'' |HTM}} < ''{{Gr|epsilon}}'' |&&}}
yields the display style HTML+CSS
f ′ (x) = d iff ∀ϵ, ∃δ s.t.
0 < | Δ x | < δ with
|
while the code
{{indent}}{{math| ''f{{sym|prime|tex}}'' (''x'') {{=}} ''d'' \text{ iff } {{sym|forall|tex}}''{{Gr|epsilon|tex}}'',\, {{sym|exists|tex}}''{{Gr|delta|tex}}'' \text{ s.t. } 0 < {{abs| {{Gr|Delta|tex}}{{thinsp|tex}}''x'' |tex}} < ''{{Gr|delta|tex}}'' \text{ with } {{abs| {{frac|''f''(''x'' {{op|+|tex}} {{Gr|Delta|tex}}{{thinsp|tex}}''x'') {{op|-|tex}} ''f''(''x'') | {{Gr|Delta|tex}}{{thinsp|tex}}''x'' |TEX}} {{op|-|tex}} ''d'' |TEX}} < ''{{Gr|epsilon|tex}}'' |$$}}
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) :=
|
or yields the display style LaTeX (with the $$
option):
Maxwell’s equations[1] (as differential equations)
The code
{{indent}}{{math| {{align|begin}} {{op|curl}}{{vec|B|b}} {{op|-}} {{frac|1|''c''|HTM}}{{sp|2}}{{partial|{{vec|E|b}}|''t''|HTM}} {{&=}} {{frac|4{{Gr|pi}}|''c''|HTM}}{{sp|1}}{{vec|J|b}} {{\\}} {{op|divergence}}{{vec|E|b}} {{&=}} 4{{Gr|pi}}{{sp|3}}''{{Gr|rho}}'' {{\\}} {{op|curl}}{{vec|E|b}} {{op|+}} {{frac|1|''c''|HTM}}{{sp|2}}{{partial|{{vec|B|b}}|''t''|HTM}} {{&=}} {{vec|0|b}} {{\\}} {{op|divergence}}{{vec|B|b}} {{&=}} 0 {{align|end}} |tex = {{align|begin|tex}} {{op|curl|tex}}{{vec|B|b|tex}} - {{frac|1|c|TEX}} {{partial|{{vec|E|b|tex}}|t|TEX}} {{&=|tex}} {{frac|4{{Gr|pi|tex}}|c|TEX}} {{vec|J|b|tex}} {{\\|tex}} {{op|divergence|tex}}{{vec|E|b|tex}} {{&=|tex}} 4{{Gr|pi|tex}} {{Gr|rho|tex}} {{\\|tex}} {{op|curl|tex}}{{vec|E|b|tex}} + {{frac|1|c|TEX}} {{partial|{{vec|B|b|tex}}|t|TEX}} {{&=|tex}} {{vec|0|b|tex}} {{\\|tex}} {{op|divergence|tex}}{{vec|B|b|tex}} {{&=|tex}} 0 {{align|end|tex}} |&&}}
yields the display style HTML+CSS (with the &&
option):
|
or yields the display style LaTeX (with the $$
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 −
|
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} |$$}}
yields the display style LaTeX
Riemann curvature tensor
The code
{{indent}}{{math|<!-- LaTeX doesn't seem to italicize lambda, so we won't! --><!-- Cumbersome code... --> <!-- -->''R''{{sup|''{{Gr|rho}}''}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|mu}}{{sp|1}}{{Gr|nu}}''}} {{=}} <!-- -->{{sym|partial}}{{sub|''{{Gr|mu}}''}} {{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|nu}}''}} {{op|-}} <!-- -->{{sym|partial}}{{sub|''{{Gr|nu}}''}} {{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|mu}}''}} {{op|+}} <!-- -->{{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|{{Gr|lambda}}{{sp|1}}''{{Gr|mu}}''}} <!-- -->{{Gr|Gamma}}{{sup|{{Gr|lambda}}}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|nu}}''}} {{op|-}} <!-- -->{{Gr|Gamma}}{{sup|''{{Gr|rho}}''}}{{sub|{{Gr|lambda}}{{sp|1}}''{{Gr|nu}}''}} <!-- -->{{Gr|Gamma}}{{sup|{{Gr|lambda}}}}{{sub|''{{Gr|sigma}}{{sp|1}}{{Gr|mu}}''}}<!-- --> |&&}}yields the display style HTML+CSS (italicized lambda
λ |
Rρσ μ ν = ∂μ Γρσ ν − ∂ν Γρσ μ + Γρλ μ Γλσ ν − Γρλ ν Γλσ μ |
while the code
{{indent}}{{math| <!-- -->{ R^{\rho} }_{\sigma \mu \nu} {{=}} \partial_{\mu} { \Gamma^{\rho} }_{\sigma \nu} - \partial_{\nu} { \Gamma^{\rho} }_{\sigma \mu} + { \Gamma^{\rho} }_{\lambda \mu} \, { \Gamma^{\lambda} }_{\sigma \nu} - { \Gamma^{\rho} }_{\lambda \nu} \, { \Gamma^{\lambda} }_{\sigma \mu}<!-- --> |$$}}
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 equationy 3 + 3py + 2q = 0 |
D = q 2 + p 3 > 0 |
y1 = u + v, y2 = −
|
where
u = 3√ − q + , v = √ q 2 + p 3 3√ − q − .√ 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 curveC |
r (t) = (x (t), y (t)) |
κ =
|
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}}. |$$}}
yields the LaTeX:
The curvature at any point of the plane curve given by is
Catalan numbers
The code
{{indent}}{{math|<!-- -->''C''{{sub|''n''}} {{=}} {{frac|1|''n'' + 1|HTM}} {{binom|2''n''|''n''|HTM}} {{=}} {{frac|(2''n'')!|(''n'' + 1)!{{sp|3}}''n''!|HTM}} {{=}} <!-- -->{{prod|''k''{{=}}2|''n''|{{frac|''n'' + ''k''|''k''|HTM}}|HTM}},{{sp|quad}}''n'' {{rel|ge}} 0.<!-- -->|&&}}
yields the display style HTML+CSS
Cn =
|
The code
{{indent}}{{math|<!-- -->''C''{{sub|''n''|TEX}} {{=}} {{frac|1|''n'' + 1|TEX}} {{binom|2''n''|''n''|TEX}} {{=}} {{frac|(2''n'')!|(''n'' + 1)!{{sp|3|tex}}''n''!|TEX}} {{=}} <!-- -->{{prod|''k''{{=}}2|''n''|{{frac|''n'' + ''k''|''k''|TEX}}|TEX}},{{sp|quad|TEX}}''n'' {{rel|ge|TEX}} 0.<!-- -->|$$}}
yields the display style LaTeX
Summations and integrals
Mellin–Barnes integral[4]
Cf. Mellin–Barnes 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) =
|
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 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 |
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 |
R − a (b; z) =
|
|
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 ofγ/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 |$$}}
yields the display style HTML+CSS (with the &&
option)
4√ z 2 + sech (π z) 2 d z
|
or yields the display style LaTeX (with the $$
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) =
|
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}}.|$$}}
yields the display style LaTeX
The code
: <math>\delta(x) = \lim_{n \to \infty} \frac{1}{2\pi} \frac{\sin [(n + \frac{1}{2}) x]}{\sin (\frac{1}{2} x)}.</math>
yields the display style LaTeX
BPP formula
The code
: {{math |{{Gr|pi}} {{=}} {{sum|''k''{{=}}0|infty|HTM}} {{sqbrack|Big l}} {{frac|1|16{{^|''k''}}|HTM}} <!-- -->{{paren|Big l}}{{frac|4|8''k'' + 1|HTM}} {{op|-}} {{frac|2|8''k'' + 4|HTM}} {{op|-}} {{frac|1|8''k'' + 5|HTM}} {{op|-}} {{frac|1|8''k'' + 6|HTM}}{{paren|Big r}} {{sqbrack|Big r}}, |tex = \pi = \sum_{k{{=}}0}^{\infty} \left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right], |&&}}
yields the display style HTML+CSS (the {{(|...|)}} template needs more work!)
-
π = ∞∑ k = 0⎡ ⎣ 1 16 k ⎛ ⎝
−4 8k + 1
−2 8k + 4
−1 8k + 5 1 8k + 6 ⎞ ⎠
,⎤ ⎦
and with the $$
option yields the display style LaTeX
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.