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docs: render equations via math code blocks
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Planeshifter committed Mar 8, 2023
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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/absf/README.md
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Expand Up @@ -28,10 +28,14 @@ The [absolute value][absolute-value] is defined as

<!-- <equation class="equation" label="eq:absolute_value" align="center" raw="|x| = \begin{cases} x & \textrm{if}\ x \geq 0 \\ -x & \textrm{if}\ x < 0\end{cases}" alt="Absolute value"> -->

<div class="equation" align="center" data-raw-text="|x| = \begin{cases} x &amp; \textrm{if}\ x \geq 0 \\ -x &amp; \textrm{if}\ x &lt; 0\end{cases}" data-equation="eq:absolute_value">
```math
|x| = \begin{cases} x & \textrm{if}\ x \geq 0 \\ -x & \textrm{if}\ x < 0\end{cases}
```

<!-- <div class="equation" align="center" data-raw-text="|x| = \begin{cases} x &amp; \textrm{if}\ x \geq 0 \\ -x &amp; \textrm{if}\ x &lt; 0\end{cases}" data-equation="eq:absolute_value">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@5b4ee1217697eb737011e0d588fddbb119806753/lib/node_modules/@stdlib/math/base/special/absf/docs/img/equation_absolute_value.svg" alt="Absolute value">
<br>
</div>
</div> -->

<!-- </equation> -->

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Expand Up @@ -28,10 +28,14 @@ The [inverse coversed cosine][inverse-coversed-cosine] is defined as

<!-- <equation class="equation" label="eq:arccovercosine" align="center" raw="\operatorname{acovercos}(\theta) = \arcsin(1+\theta)" alt="Inverse coversed cosine."> -->

<div class="equation" align="center" data-raw-text="\operatorname{acovercos}(\theta) = \arcsin(1+\theta)" data-equation="eq:arccovercosine">
```math
\operatorname{acovercos}(\theta) = \arcsin(1+\theta)
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{acovercos}(\theta) = \arcsin(1+\theta)" data-equation="eq:arccovercosine">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/acovercos/docs/img/equation_arccovercosine.svg" alt="Inverse coversed cosine.">
<br>
</div>
</div> -->

<!-- </equation> -->

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Expand Up @@ -28,10 +28,14 @@ The [inverse coversed sine][inverse-coversed-sine] is defined as

<!-- <equation class="equation" label="eq:arccoversine" align="center" raw="\operatorname{acoversin}(\theta) = \arcsin(1-\theta)" alt="Inverse coversed sine."> -->

<div class="equation" align="center" data-raw-text="\operatorname{acoversin}(\theta) = \arcsin(1-\theta)" data-equation="eq:arccoversine">
```math
\operatorname{acoversin}(\theta) = \arcsin(1-\theta)
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{acoversin}(\theta) = \arcsin(1-\theta)" data-equation="eq:arccoversine">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/acoversin/docs/img/equation_arccoversine.svg" alt="Inverse coversed sine.">
<br>
</div>
</div> -->

<!-- </equation> -->

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Expand Up @@ -28,10 +28,14 @@ The [inverse half-value versed cosine][archavercosine] is defined as

<!-- <equation class="equation" label="eq:archavercosine" align="center" raw="\operatorname{ahavercos}(\theta) = 2 \cdot \arccos(\sqrt{\theta})" alt="Inverse half-value versed cosine."> -->

<div class="equation" align="center" data-raw-text="\operatorname{ahavercos}(\theta) = 2 \cdot \arccos(\sqrt{\theta})" data-equation="eq:archavercosine">
```math
\operatorname{ahavercos}(\theta) = 2 \cdot \arccos(\sqrt{\theta})
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{ahavercos}(\theta) = 2 \cdot \arccos(\sqrt{\theta})" data-equation="eq:archavercosine">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/ahavercos/docs/img/equation_archavercosine.svg" alt="Inverse half-value versed cosine.">
<br>
</div>
</div> -->

<!-- </equation> -->

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Expand Up @@ -28,10 +28,14 @@ The [inverse half-value versed sine][archaversine] is defined as

<!-- <equation class="equation" label="eq:archaversine" align="center" raw="\operatorname{ahaversin}(\theta) = 2 \cdot \arcsin(\sqrt{\theta})" alt="Inverse half-value versed sine."> -->

<div class="equation" align="center" data-raw-text="\operatorname{ahaversin}(\theta) = 2 \cdot \arcsin(\sqrt{\theta})" data-equation="eq:archaversine">
```math
\operatorname{ahaversin}(\theta) = 2 \cdot \arcsin(\sqrt{\theta})
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{ahaversin}(\theta) = 2 \cdot \arcsin(\sqrt{\theta})" data-equation="eq:archaversine">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/ahaversin/docs/img/equation_archaversine.svg" alt="Inverse half-value versed sine.">
<br>
</div>
</div> -->

<!-- </equation> -->

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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/avercos/README.md
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Expand Up @@ -28,10 +28,14 @@ The [inverse versed cosine][inverse-versed-cosine] is defined as

<!-- <equation class="equation" label="eq:arcvercosine" align="center" raw="\operatorname{avercos}(\theta) = \arccos(1+\theta)" alt="Inverse versed cosine."> -->

<div class="equation" align="center" data-raw-text="\operatorname{avercos}(\theta) = \arccos(1+\theta)" data-equation="eq:arcvercosine">
```math
\operatorname{avercos}(\theta) = \arccos(1+\theta)
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{avercos}(\theta) = \arccos(1+\theta)" data-equation="eq:arcvercosine">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/avercos/docs/img/equation_arcvercosine.svg" alt="Inverse versed cosine.">
<br>
</div>
</div> -->

<!-- </equation> -->

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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/aversin/README.md
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Expand Up @@ -28,10 +28,14 @@ The [inverse versed sine][inverse-versed-sine] is defined as

<!-- <equation class="equation" label="eq:arcversine" align="center" raw="\operatorname{aversin}(\theta) = \arccos(1-\theta)" alt="Inverse versed sine."> -->

<div class="equation" align="center" data-raw-text="\operatorname{aversin}(\theta) = \arccos(1-\theta)" data-equation="eq:arcversine">
```math
\operatorname{aversin}(\theta) = \arccos(1-\theta)
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{aversin}(\theta) = \arccos(1-\theta)" data-equation="eq:arcversine">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/aversin/docs/img/equation_arcversine.svg" alt="Inverse versed sine.">
<br>
</div>
</div> -->

<!-- </equation> -->

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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/besselj0/README.md
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Expand Up @@ -28,10 +28,14 @@ The [Bessel function of the first kind][bessel-first-kind] of order zero is defi

<!-- <equation class="equation" label="eq:bessel_first_kind_order_zero" align="center" raw="J_0 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{- i x \sin(\tau)} \,d\tau." alt="Bessel function of the first kind of order zero"> -->

<div class="equation" align="center" data-raw-text="J_0 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{- i x \sin(\tau)} \,d\tau." data-equation="eq:bessel_first_kind_order_zero">
```math
J_0 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{- i x \sin(\tau)} \,d\tau.
```

<!-- <div class="equation" align="center" data-raw-text="J_0 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{- i x \sin(\tau)} \,d\tau." data-equation="eq:bessel_first_kind_order_zero">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/besselj0/docs/img/equation_bessel_first_kind_order_zero.svg" alt="Bessel function of the first kind of order zero">
<br>
</div>
</div> -->

<!-- </equation> -->

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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/besselj1/README.md
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Expand Up @@ -28,10 +28,14 @@ The [Bessel function of the first kind][bessel-first-kind] of order one is defin

<!-- <equation class="equation" label="eq:bessel_first_kind_order_one" align="center" raw="J_1 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(\tau - x \sin(\tau))} \,d\tau" alt="Bessel function of the first kind of order one"> -->

<div class="equation" align="center" data-raw-text="J_1 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(\tau - x \sin(\tau))} \,d\tau" data-equation="eq:bessel_first_kind_order_one">
```math
J_1 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(\tau - x \sin(\tau))} \,d\tau
```

<!-- <div class="equation" align="center" data-raw-text="J_1 (x) = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(\tau - x \sin(\tau))} \,d\tau" data-equation="eq:bessel_first_kind_order_one">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/besselj1/docs/img/equation_bessel_first_kind_order_one.svg" alt="Bessel function of the first kind of order one">
<br>
</div>
</div> -->

<!-- </equation> -->

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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/bessely0/README.md
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Expand Up @@ -28,10 +28,14 @@ The [Bessel function of the second kind][bessel-second-kind] of order zero is de

<!-- <equation class="equation" label="eq:bessel_second_kind_order_zero" align="center" raw="Y_0(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta) \, d\theta -\frac{2}{\pi} \int_0^\infty e^{-x \sinh t} \, dt." alt="Bessel function of the second kind of order zero"> -->

<div class="equation" align="center" data-raw-text="Y_0(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta) \, d\theta -\frac{2}{\pi} \int_0^\infty e^{-x \sinh t} \, dt." data-equation="eq:bessel_second_kind_order_zero">
```math
Y_0(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta) \, d\theta -\frac{2}{\pi} \int_0^\infty e^{-x \sinh t} \, dt.
```

<!-- <div class="equation" align="center" data-raw-text="Y_0(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta) \, d\theta -\frac{2}{\pi} \int_0^\infty e^{-x \sinh t} \, dt." data-equation="eq:bessel_second_kind_order_zero">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/bessely0/docs/img/equation_bessel_second_kind_order_zero.svg" alt="Bessel function of the second kind of order zero">
<br>
</div>
</div> -->

<!-- </equation> -->

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8 changes: 6 additions & 2 deletions lib/node_modules/@stdlib/math/base/special/bessely1/README.md
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Expand Up @@ -28,10 +28,14 @@ The [Bessel function of the second kind][bessel-second-kind] of order one is def

<!-- <equation class="equation" label="eq:bessel_second_kind_order_one" align="center" raw="Y_1(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - \theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left[ e^t - e^{-t} \right] e^{-x \sinh t} \, dt" alt="Bessel function of the second kind of order one"> -->

<div class="equation" align="center" data-raw-text="Y_1(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - \theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left[ e^t - e^{-t} \right] e^{-x \sinh t} \, dt" data-equation="eq:bessel_second_kind_order_one">
```math
Y_1(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - \theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left[ e^t - e^{-t} \right] e^{-x \sinh t} \, dt
```

<!-- <div class="equation" align="center" data-raw-text="Y_1(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - \theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left[ e^t - e^{-t} \right] e^{-x \sinh t} \, dt" data-equation="eq:bessel_second_kind_order_one">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/bessely1/docs/img/equation_bessel_second_kind_order_one.svg" alt="Bessel function of the second kind of order one">
<br>
</div>
</div> -->

<!-- </equation> -->

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16 changes: 12 additions & 4 deletions lib/node_modules/@stdlib/math/base/special/beta/README.md
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Expand Up @@ -28,21 +28,29 @@ The [beta function][beta-function], also called the Euler integral, is defined a

<!-- <equation class="equation" label="eq:beta_function" align="center" raw="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" alt="Equation for the beta function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" data-equation="eq:beta_function">
```math
\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" data-equation="eq:beta_function">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/beta/docs/img/equation_beta_function.svg" alt="Equation for the beta function.">
<br>
</div>
</div> -->

<!-- </equation> -->

The [beta function][beta-function] is related to the [Gamma function][gamma-function] via the following equation

<!-- <equation class="equation" label="eq:beta_function2" align="center" raw="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" alt="Beta function expressed in terms of the Gamma function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" data-equation="eq:beta_function2">
```math
\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" data-equation="eq:beta_function2">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/beta/docs/img/equation_beta_function2.svg" alt="Beta function expressed in terms of the Gamma function.">
<br>
</div>
</div> -->

<!-- </equation> -->

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16 changes: 12 additions & 4 deletions lib/node_modules/@stdlib/math/base/special/betaln/README.md
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Expand Up @@ -28,21 +28,29 @@ The [beta function][beta-function], also called the Euler integral, is defined a

<!-- <equation class="equation" label="eq:beta_function" align="center" raw="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" alt="Equation for the beta function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" data-equation="eq:beta_function">
```math
\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" data-equation="eq:beta_function">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/betaln/docs/img/equation_beta_function.svg" alt="Equation for the beta function.">
<br>
</div>
</div> -->

<!-- </equation> -->

The [beta function][beta-function] is related to the [gamma function][gamma-function] via the following equation

<!-- <equation class="equation" label="eq:beta_function2" align="center" raw="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" alt="Beta function expressed in terms of the Gamma function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" data-equation="eq:beta_function2">
```math
\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!
```

<!-- <div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" data-equation="eq:beta_function2">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/betaln/docs/img/equation_beta_function2.svg" alt="Beta function expressed in terms of the Gamma function.">
<br>
</div>
</div> -->

<!-- </equation> -->

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16 changes: 12 additions & 4 deletions lib/node_modules/@stdlib/math/base/special/binet/README.md
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Expand Up @@ -28,21 +28,29 @@ limitations under the License.

<!-- <equation class="equation" label="eq:binets_formula" align="center" raw="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" alt="Binet's formula"> -->

<div class="equation" align="center" data-raw-text="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" data-equation="eq:binets_formula">
```math
F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}
```

<!-- <div class="equation" align="center" data-raw-text="F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}" data-equation="eq:binets_formula">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binet/docs/img/equation_binets_formula.svg" alt="Binet's formula">
<br>
</div>
</div> -->

<!-- </equation> -->

where `φ` is the [golden ratio][golden-ratio] and `ψ` is `1 - φ`. To extend [Fibonacci numbers][fibonacci-number] to real numbers, we may express [Binet's formula][fibonacci-number] as

<!-- <equation class="equation" label="eq:binets_formula_real_numbers" align="center" raw="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" alt="Binet's formula extended to real numbers."> -->

<div class="equation" align="center" data-raw-text="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" data-equation="eq:binets_formula_real_numbers">
```math
F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}
```

<!-- <div class="equation" align="center" data-raw-text="F_x = \frac{\varphi^x - \varphi^{-x} \cdot \cos(\pi x)}{\sqrt{5}}" data-equation="eq:binets_formula_real_numbers">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binet/docs/img/equation_binets_formula_real_numbers.svg" alt="Binet's formula extended to real numbers.">
<br>
</div>
</div> -->

<!-- </equation> -->

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16 changes: 12 additions & 4 deletions lib/node_modules/@stdlib/math/base/special/binomcoef/README.md
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Expand Up @@ -28,21 +28,29 @@ The [binomial coefficient][binomial-coefficient] of two nonnegative integers `n`

<!-- <equation class="equation" label="eq:binomial_coefficient" align="center" raw="\binom {n}{k} = \frac{n!}{k!\,(n-k)!} \quad \text{for }\ 0\leq k\leq n" alt="Factorial formula for the Binomial coefficient."> -->

<div class="equation" align="center" data-raw-text="\binom {n}{k} = \frac{n!}{k!\,(n-k)!} \quad \text{for }\ 0\leq k\leq n" data-equation="eq:binomial_coefficient">
```math
\binom {n}{k} = \frac{n!}{k!\,(n-k)!} \quad \text{for }\ 0\leq k\leq n
```

<!-- <div class="equation" align="center" data-raw-text="\binom {n}{k} = \frac{n!}{k!\,(n-k)!} \quad \text{for }\ 0\leq k\leq n" data-equation="eq:binomial_coefficient">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binomcoef/docs/img/equation_binomial_coefficient.svg" alt="Factorial formula for the Binomial coefficient.">
<br>
</div>
</div> -->

<!-- </equation> -->

The [binomial coefficient][binomial-coefficient] can be generalized to negative integers `n` as follows:

<!-- <equation class="equation" label="eq:binomial_coefficient_negative_integers" align="center" raw="\binom {-n}{k} = (-1)^{k} \binom{n + k - 1}{k} = (-1)^{k} \left(\!\!{\binom {n}{k}}\!\!\right)" alt="Generalization of the binomial coefficient to negative n."> -->

<div class="equation" align="center" data-raw-text="\binom {-n}{k} = (-1)^{k} \binom{n + k - 1}{k} = (-1)^{k} \left(\!\!{\binom {n}{k}}\!\!\right)" data-equation="eq:binomial_coefficient_negative_integers">
```math
\binom {-n}{k} = (-1)^{k} \binom{n + k - 1}{k} = (-1)^{k} \left(\!\!{\binom {n}{k}}\!\!\right)
```

<!-- <div class="equation" align="center" data-raw-text="\binom {-n}{k} = (-1)^{k} \binom{n + k - 1}{k} = (-1)^{k} \left(\!\!{\binom {n}{k}}\!\!\right)" data-equation="eq:binomial_coefficient_negative_integers">
<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/binomcoef/docs/img/equation_binomial_coefficient_negative_integers.svg" alt="Generalization of the binomial coefficient to negative n.">
<br>
</div>
</div> -->

<!-- </equation> -->

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