added volume of d-ball

data/Integrals/Periods/Complete_elliptic_integral_of_the_second_kind_E/collection.yaml

@@ -23,7 +23,7 @@ Comments:
   
 Formulas: 
   formula-sin: >
-    $E(m) = \int_0^{\pi/2} \sqrt{1-m\sin^2\theta}} d\theta$.
+    $E(m) = \int_0^{\pi/2} \sqrt{1-m\sin^2\theta} d\theta$.
 
 Programs: 
 

data/Integrals/Periods/Volume_of_the_unit_ball/collection.yaml

@@ -0,0 +1,52 @@
+ID: INPUT{id.yaml}
+
+Title: >
+  Volume of the $d$-dimensional unit ball
+
+Definition: >
+  $B_d = \text{vol}(\{x\in\mathbb{R}^d: |x|\leq 1\})$ is 
+  the volume of the unit ball in $d$-dimensional Euclidean space.
+
+Parameters:
+  d:
+    display: $d$
+    type: Z
+    constraints: $d \geq 0$
+
+Comments:
+
+Formulas: 
+  formula-gamma: >
+    $B_d = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$.
+  formula-only-gamma: >
+    $B_d = \frac{2^d\Gamma(1+1/2)^d}{\Gamma(1+d/2)}$.
+  formula-sphere: >
+    $dB_d = S_{d-1}$, where $S_{d-1}$ denotes the 
+    $(d-1)$-dimensional volume of the unit sphere in $\mathbb{R}^d$.
+
+Programs: 
+
+References:
+
+Links:
+  Wiki:
+    title: "Wikipedia: Volume of an n-ball"
+    url: https://en.wikipedia.org/wiki/Volume_of_an_n-ball
+
+Similar tables:
+
+Keywords:
+
+Tags:
+- elementary geometry
+- integral
+- period
+
+Data properties:
+  type: R
+  complete: no
+
+Display properties:
+  number-header: $B_d$
+
+Numbers: INPUT{numbers.yaml}

data/Integrals/Periods/Volume_of_the_unit_ball/generate.sage

@@ -0,0 +1,29 @@
+import yaml
+import os
+from utils.utils import numbers_to_yaml
+from utils.utils import real_interval_to_sage_string
+
+
+path = 'data/Integrals/Periods/Volume_of_the_unit_ball/'
+
+prec10 = 100 #relative precision in base 10
+
+d_range = [0..500]
+
+RIFprec = RealIntervalField(prec10 * 3.4 * 2)
+
+numbers = {}
+for d in d_range:
+	print("d:",d)
+
+	number = RIFprec(pi^(d/2)/gamma(d/2+1))
+
+	number_str = real_interval_to_sage_string(
+		RIFprec(number),
+		max_digits = prec10,
+	).replace('?','')
+
+	numbers[str(d)] = number_str
+
+filename = os.path.join(path, 'numbers.yaml')
+yaml.dump(numbers, stream = open(filename, 'w'), sort_keys = False)