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tmp/tmpd3v5h117/{from.md → to.md} RENAMED
@@ -26,14 +26,12 @@ long double assoc_laguerrel(unsigned n, unsigned m, long double x);
26
  ```
27
 
28
  *Effects:* These functions compute the associated Laguerre polynomials
29
  of their respective arguments `n`, `m`, and `x`.
30
 
31
- *Returns:* $$\mathsf{L}_n^m(x) =
32
- (-1)^m \frac{\mathsf{d} ^ m}{\mathsf{d}x ^ m} \, \mathsf{L}_{n+m}(x)
33
- \text{ ,\quad for $x \ge 0$,}$$ where n is `n`, m is `m`, and x is
34
- `x`.
35
 
36
  *Remarks:* The effect of calling each of these functions is
37
  *implementation-defined* if `n >= 128` or if `m >= 128`.
38
 
39
  #### Associated Legendre functions <a id="sf.cmath.assoc.legendre">[[sf.cmath.assoc.legendre]]</a>
@@ -45,14 +43,12 @@ long double assoc_legendrel(unsigned l, unsigned m, long double x);
45
  ```
46
 
47
  *Effects:* These functions compute the associated Legendre functions of
48
  their respective arguments `l`, `m`, and `x`.
49
 
50
- *Returns:* $$\mathsf{P}_\ell^m(x) = (1 - x^2) ^ {m/2} \:
51
- \frac{\mathsf{d} ^ m}{\mathsf{d}x ^ m} \, \mathsf{P}_\ell(x)
52
- \text{ ,\quad for $|x| \le 1$,}$$ where l is `l`, m is `m`, and x is
53
- `x`.
54
 
55
  *Remarks:* The effect of calling each of these functions is
56
  *implementation-defined* if `l >= 128`.
57
 
58
  #### Beta function <a id="sf.cmath.beta">[[sf.cmath.beta]]</a>
@@ -64,13 +60,11 @@ long double betal(long double x, long double y);
64
  ```
65
 
66
  *Effects:* These functions compute the beta function of their respective
67
  arguments `x` and `y`.
68
 
69
- *Returns:*
70
- $$\mathsf{B}(x, y) = \frac{\Gamma(x) \, \Gamma(y)}{\Gamma(x + y)}
71
- \text{ ,\quad for $x > 0$,\, $y > 0$,}$$ where x is `x` and y is `y`.
72
 
73
  #### Complete elliptic integral of the first kind <a id="sf.cmath.comp.ellint.1">[[sf.cmath.comp.ellint.1]]</a>
74
 
75
  ``` cpp
76
  floating-point-type comp_ellint_1(floating-point-type k);
@@ -79,13 +73,11 @@ long double comp_ellint_1l(long double k);
79
  ```
80
 
81
  *Effects:* These functions compute the complete elliptic integral of the
82
  first kind of their respective arguments `k`.
83
 
84
- *Returns:*
85
- $$\mathsf{K}(k) = \mathsf{F}(k, \pi / 2) \text{ ,\quad for $|k| \le 1$,}$$
86
- where k is `k`.
87
 
88
  See also [[sf.cmath.ellint.1]].
89
 
90
  #### Complete elliptic integral of the second kind <a id="sf.cmath.comp.ellint.2">[[sf.cmath.comp.ellint.2]]</a>
91
 
@@ -96,13 +88,11 @@ long double comp_ellint_2l(long double k);
96
  ```
97
 
98
  *Effects:* These functions compute the complete elliptic integral of the
99
  second kind of their respective arguments `k`.
100
 
101
- *Returns:*
102
- $$\mathsf{E}(k) = \mathsf{E}(k, \pi / 2) \text{ ,\quad for $|k| \le 1$,}$$
103
- where k is `k`.
104
 
105
  See also [[sf.cmath.ellint.2]].
106
 
107
  #### Complete elliptic integral of the third kind <a id="sf.cmath.comp.ellint.3">[[sf.cmath.comp.ellint.3]]</a>
108
 
@@ -113,13 +103,12 @@ long double comp_ellint_3l(long double k, long double nu);
113
  ```
114
 
115
  *Effects:* These functions compute the complete elliptic integral of the
116
  third kind of their respective arguments `k` and `nu`.
117
 
118
- *Returns:*
119
- $$\mathsf{\Pi}(\nu, k) = \mathsf{\Pi}(\nu, k, \pi / 2) \text{ ,\quad for $|k| \le 1$,}$$
120
- where k is `k` and $\nu$ is `nu`.
121
 
122
  See also [[sf.cmath.ellint.3]].
123
 
124
  #### Regular modified cylindrical Bessel functions <a id="sf.cmath.cyl.bessel.i">[[sf.cmath.cyl.bessel.i]]</a>
125
 
@@ -130,14 +119,12 @@ long double cyl_bessel_il(long double nu, long double x);
130
  ```
131
 
132
  *Effects:* These functions compute the regular modified cylindrical
133
  Bessel functions of their respective arguments `nu` and `x`.
134
 
135
- *Returns:* $$\mathsf{I}_\nu(x) =
136
- i^{-\nu} \mathsf{J}_\nu(ix) =
137
- \sum_{k=0}^\infty \frac{(x/2)^{\nu+2k}}{k! \: \Gamma(\nu+k+1)}
138
- \text{ ,\quad for $x \ge 0$,}$$ where $\nu$ is `nu` and x is `x`.
139
 
140
  *Remarks:* The effect of calling each of these functions is
141
  *implementation-defined* if `nu >= 128`.
142
 
143
  See also [[sf.cmath.cyl.bessel.j]].
@@ -151,13 +138,12 @@ long double cyl_bessel_jl(long double nu, long double x);
151
  ```
152
 
153
  *Effects:* These functions compute the cylindrical Bessel functions of
154
  the first kind of their respective arguments `nu` and `x`.
155
 
156
- *Returns:* $$\mathsf{J}_\nu(x) =
157
- \sum_{k=0}^\infty \frac{(-1)^k (x/2)^{\nu+2k}}{k! \: \Gamma(\nu+k+1)}
158
- \text{ ,\quad for $x \ge 0$,}$$ where $\nu$ is `nu` and x is `x`.
159
 
160
  *Remarks:* The effect of calling each of these functions is
161
  *implementation-defined* if `nu >= 128`.
162
 
163
  #### Irregular modified cylindrical Bessel functions <a id="sf.cmath.cyl.bessel.k">[[sf.cmath.cyl.bessel.k]]</a>
@@ -169,32 +155,12 @@ long double cyl_bessel_kl(long double nu, long double x);
169
  ```
170
 
171
  *Effects:* These functions compute the irregular modified cylindrical
172
  Bessel functions of their respective arguments `nu` and `x`.
173
 
174
- *Returns:* $$%
175
- \mathsf{K}_\nu(x) =
176
- (\pi/2)i^{\nu+1} ( \mathsf{J}_\nu(ix)
177
- + i \mathsf{N}_\nu(ix)
178
- )
179
- =
180
- \left\{
181
- \begin{array}{cl}
182
- \displaystyle
183
- \frac{\pi}{2}
184
- \frac{\mathsf{I}_{-\nu}(x) - \mathsf{I}_{\nu}(x)}
185
- {\sin \nu\pi },
186
- & \mbox{for $x \ge 0$ and non-integral $\nu$}
187
- \\
188
- \\
189
- \displaystyle
190
- \frac{\pi}{2}
191
- \lim_{\mu \rightarrow \nu} \frac{\mathsf{I}_{-\mu}(x) - \mathsf{I}_{\mu}(x)}
192
- {\sin \mu\pi },
193
- & \mbox{for $x \ge 0$ and integral $\nu$}
194
- \end{array}
195
- \right.$$ where $\nu$ is `nu` and x is `x`.
196
 
197
  *Remarks:* The effect of calling each of these functions is
198
  *implementation-defined* if `nu >= 128`.
199
 
200
  See also [[sf.cmath.cyl.bessel.i]], [[sf.cmath.cyl.bessel.j]],
@@ -210,26 +176,12 @@ long double cyl_neumannl(long double nu, long double x);
210
 
211
  *Effects:* These functions compute the cylindrical Neumann functions,
212
  also known as the cylindrical Bessel functions of the second kind, of
213
  their respective arguments `nu` and `x`.
214
 
215
- *Returns:* $$%
216
- \mathsf{N}_\nu(x) =
217
- \left\{
218
- \begin{array}{cl}
219
- \displaystyle
220
- \frac{\mathsf{J}_\nu(x) \cos \nu\pi - \mathsf{J}_{-\nu}(x)}
221
- {\sin \nu\pi },
222
- & \mbox{for $x \ge 0$ and non-integral $\nu$}
223
- \\
224
- \\
225
- \displaystyle
226
- \lim_{\mu \rightarrow \nu} \frac{\mathsf{J}_\mu(x) \cos \mu\pi - \mathsf{J}_{-\mu}(x)}
227
- {\sin \mu\pi },
228
- & \mbox{for $x \ge 0$ and integral $\nu$}
229
- \end{array}
230
- \right.$$ where $\nu$ is `nu` and x is `x`.
231
 
232
  *Remarks:* The effect of calling each of these functions is
233
  *implementation-defined* if `nu >= 128`.
234
 
235
  See also [[sf.cmath.cyl.bessel.j]].
@@ -244,13 +196,11 @@ long double ellint_1l(long double k, long double phi);
244
 
245
  *Effects:* These functions compute the incomplete elliptic integral of
246
  the first kind of their respective arguments `k` and `phi` (`phi`
247
  measured in radians).
248
 
249
- *Returns:* $$\mathsf{F}(k, \phi) =
250
- \int_0^\phi \! \frac{\mathsf{d}\theta}{\sqrt{1 - k^2 \sin^2 \theta}}
251
- \text{ ,\quad for $|k| \le 1$,}$$ where k is `k` and φ is `phi`.
252
 
253
  #### Incomplete elliptic integral of the second kind <a id="sf.cmath.ellint.2">[[sf.cmath.ellint.2]]</a>
254
 
255
  ``` cpp
256
  floating-point-type ellint_2(floating-point-type k, floating-point-type phi);
@@ -260,13 +210,11 @@ long double ellint_2l(long double k, long double phi);
260
 
261
  *Effects:* These functions compute the incomplete elliptic integral of
262
  the second kind of their respective arguments `k` and `phi` (`phi`
263
  measured in radians).
264
 
265
- *Returns:*
266
- $$\mathsf{E}(k, \phi) = \int_0^\phi \! \sqrt{1 - k^2 \sin^2 \theta} \, \mathsf{d}\theta
267
- \text{ ,\quad for $|k| \le 1$,}$$ where k is `k` and φ is `phi`.
268
 
269
  #### Incomplete elliptic integral of the third kind <a id="sf.cmath.ellint.3">[[sf.cmath.ellint.3]]</a>
270
 
271
  ``` cpp
272
  floating-point-type ellint_3(floating-point-type k, floating-point-type nu,
@@ -277,13 +225,12 @@ long double ellint_3l(long double k, long double nu, long double phi);
277
 
278
  *Effects:* These functions compute the incomplete elliptic integral of
279
  the third kind of their respective arguments `k`, `nu`, and `phi` (`phi`
280
  measured in radians).
281
 
282
- *Returns:* $$\mathsf{\Pi}(\nu, k, \phi) = \int_0^\phi \!
283
- \frac{ \mathsf{d}\theta }{ (1 - \nu \, \sin^2 \theta) \sqrt{1 - k^2 \sin^2 \theta} } \text{ ,\quad for $|k| \le 1$,}$$
284
- where $\nu$ is `nu`, k is `k`, and φ is `phi`.
285
 
286
  #### Exponential integral <a id="sf.cmath.expint">[[sf.cmath.expint]]</a>
287
 
288
  ``` cpp
289
  floating-point-type expint(floating-point-type x);
@@ -292,15 +239,11 @@ long double expintl(long double x);
292
  ```
293
 
294
  *Effects:* These functions compute the exponential integral of their
295
  respective arguments `x`.
296
 
297
- *Returns:* $$%
298
- \mathsf{Ei}(x) =
299
- - \int_{-x}^\infty \frac{e^{-t}}
300
- {t } \, \mathsf{d}t
301
- \;$$ where x is `x`.
302
 
303
  #### Hermite polynomials <a id="sf.cmath.hermite">[[sf.cmath.hermite]]</a>
304
 
305
  ``` cpp
306
  floating-point-type hermite(unsigned n, floating-point-type x);
@@ -309,15 +252,11 @@ long double hermitel(unsigned n, long double x);
309
  ```
310
 
311
  *Effects:* These functions compute the Hermite polynomials of their
312
  respective arguments `n` and `x`.
313
 
314
- *Returns:* $$%
315
- \mathsf{H}_n(x) =
316
- (-1)^n e^{x^2} \frac{ \mathsf{d} ^n}
317
- { \mathsf{d}x^n} \, e^{-x^2}
318
- \;$$ where n is `n` and x is `x`.
319
 
320
  *Remarks:* The effect of calling each of these functions is
321
  *implementation-defined* if `n >= 128`.
322
 
323
  #### Laguerre polynomials <a id="sf.cmath.laguerre">[[sf.cmath.laguerre]]</a>
@@ -329,13 +268,11 @@ long double laguerrel(unsigned n, long double x);
329
  ```
330
 
331
  *Effects:* These functions compute the Laguerre polynomials of their
332
  respective arguments `n` and `x`.
333
 
334
- *Returns:* $$\mathsf{L}_n(x) =
335
- \frac{e^x}{n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} \, (x^n e^{-x})
336
- \text{ ,\quad for $x \ge 0$,}$$ where n is `n` and x is `x`.
337
 
338
  *Remarks:* The effect of calling each of these functions is
339
  *implementation-defined* if `n >= 128`.
340
 
341
  #### Legendre polynomials <a id="sf.cmath.legendre">[[sf.cmath.legendre]]</a>
@@ -347,14 +284,11 @@ long double legendrel(unsigned l, long double x);
347
  ```
348
 
349
  *Effects:* These functions compute the Legendre polynomials of their
350
  respective arguments `l` and `x`.
351
 
352
- *Returns:* $$\mathsf{P}_\ell(x) =
353
- \frac{1}{2^\ell \, \ell!}
354
- \frac{\mathsf{d}^\ell}{\mathsf{d}x^\ell} \, (x^2 - 1) ^ \ell
355
- \text{ ,\quad for $|x| \le 1$,}$$ where l is `l` and x is `x`.
356
 
357
  *Remarks:* The effect of calling each of these functions is
358
  *implementation-defined* if `l >= 128`.
359
 
360
  #### Riemann zeta function <a id="sf.cmath.riemann.zeta">[[sf.cmath.riemann.zeta]]</a>
@@ -366,32 +300,11 @@ long double riemann_zetal(long double x);
366
  ```
367
 
368
  *Effects:* These functions compute the Riemann zeta function of their
369
  respective arguments `x`.
370
 
371
- *Returns:* $$%
372
- \mathsf{\zeta}(x) =
373
- \left\{
374
- \begin{array}{cl}
375
- \displaystyle
376
- \sum_{k=1}^\infty k^{-x},
377
- & \mbox{for $x > 1$}
378
- \\
379
- \\
380
- \displaystyle
381
- \frac{1}
382
- {1 - 2^{1-x}}
383
- \sum_{k=1}^\infty (-1)^{k-1} k^{-x},
384
- & \mbox{for $0 \le x \le 1$}
385
- \\
386
- \\
387
- \displaystyle
388
- 2^x \pi^{x-1} \sin(\frac{\pi x}{2}) \, \Gamma(1-x) \, \zeta(1-x),
389
- & \mbox{for $x < 0$}
390
- \end{array}
391
- \right.
392
- \;$$ where x is `x`.
393
 
394
  #### Spherical Bessel functions of the first kind <a id="sf.cmath.sph.bessel">[[sf.cmath.sph.bessel]]</a>
395
 
396
  ``` cpp
397
  floating-point-type sph_bessel(unsigned n, floating-point-type x);
@@ -400,13 +313,11 @@ long double sph_bessell(unsigned n, long double x);
400
  ```
401
 
402
  *Effects:* These functions compute the spherical Bessel functions of the
403
  first kind of their respective arguments `n` and `x`.
404
 
405
- *Returns:*
406
- $$\mathsf{j}_n(x) = (\pi/2x)^{1\!/\!2} \mathsf{J}_{n + 1\!/\!2}(x) \text{ ,\quad for $x \ge 0$,}$$
407
- where n is `n` and x is `x`.
408
 
409
  *Remarks:* The effect of calling each of these functions is
410
  *implementation-defined* if `n >= 128`.
411
 
412
  See also [[sf.cmath.cyl.bessel.j]].
@@ -421,16 +332,12 @@ long double sph_legendrel(unsigned l, unsigned m, long double theta);
421
 
422
  *Effects:* These functions compute the spherical associated Legendre
423
  functions of their respective arguments `l`, `m`, and `theta` (`theta`
424
  measured in radians).
425
 
426
- *Returns:* $$\mathsf{Y}_\ell^m(\theta, 0)$$ where
427
- $$\mathsf{Y}_\ell^m(\theta, \phi) =
428
- (-1)^m \left[\frac{(2 \ell + 1)}{4 \pi} \frac{(\ell - m)!}{(\ell + m)!}\right]^{1/2}
429
- \mathsf{P}_\ell^m (\cos\theta) e^{i m \phi}
430
- \text{ ,\quad for $|m| \le \ell$,}$$ and l is `l`, m is `m`, and θ
431
- is `theta`.
432
 
433
  *Remarks:* The effect of calling each of these functions is
434
  *implementation-defined* if `l >= 128`.
435
 
436
  See also [[sf.cmath.assoc.legendre]].
@@ -445,13 +352,11 @@ long double sph_neumannl(unsigned n, long double x);
445
 
446
  *Effects:* These functions compute the spherical Neumann functions, also
447
  known as the spherical Bessel functions of the second kind, of their
448
  respective arguments `n` and `x`.
449
 
450
- *Returns:*
451
- $$\mathsf{n}_n(x) = (\pi/2x)^{1\!/\!2} \mathsf{N}_{n + 1\!/\!2}(x)
452
- \text{ ,\quad for $x \ge 0$,}$$ where n is `n` and x is `x`.
453
 
454
  *Remarks:* The effect of calling each of these functions is
455
  *implementation-defined* if `n >= 128`.
456
 
457
  See also [[sf.cmath.cyl.neumann]].
 
26
  ```
27
 
28
  *Effects:* These functions compute the associated Laguerre polynomials
29
  of their respective arguments `n`, `m`, and `x`.
30
 
31
+ *Returns:* Lₙᵐ(x), where Lₙᵐ is given by , Lₙ₊ₘ is given by , n is `n`,
32
+ m is `m`, and x is `x`.
 
 
33
 
34
  *Remarks:* The effect of calling each of these functions is
35
  *implementation-defined* if `n >= 128` or if `m >= 128`.
36
 
37
  #### Associated Legendre functions <a id="sf.cmath.assoc.legendre">[[sf.cmath.assoc.legendre]]</a>
 
43
  ```
44
 
45
  *Effects:* These functions compute the associated Legendre functions of
46
  their respective arguments `l`, `m`, and `x`.
47
 
48
+ *Returns:* P_ℓ^m(x), where P_ℓ^m is given by , P_ℓ is given by , ℓ is
49
+ `l`, m is `m`, and x is `x`.
 
 
50
 
51
  *Remarks:* The effect of calling each of these functions is
52
  *implementation-defined* if `l >= 128`.
53
 
54
  #### Beta function <a id="sf.cmath.beta">[[sf.cmath.beta]]</a>
 
60
  ```
61
 
62
  *Effects:* These functions compute the beta function of their respective
63
  arguments `x` and `y`.
64
 
65
+ *Returns:* B(x, y), where B is given by , x is `x` and y is `y`.
 
 
66
 
67
  #### Complete elliptic integral of the first kind <a id="sf.cmath.comp.ellint.1">[[sf.cmath.comp.ellint.1]]</a>
68
 
69
  ``` cpp
70
  floating-point-type comp_ellint_1(floating-point-type k);
 
73
  ```
74
 
75
  *Effects:* These functions compute the complete elliptic integral of the
76
  first kind of their respective arguments `k`.
77
 
78
+ *Returns:* K(k), where K is given by and k is `k`.
 
 
79
 
80
  See also [[sf.cmath.ellint.1]].
81
 
82
  #### Complete elliptic integral of the second kind <a id="sf.cmath.comp.ellint.2">[[sf.cmath.comp.ellint.2]]</a>
83
 
 
88
  ```
89
 
90
  *Effects:* These functions compute the complete elliptic integral of the
91
  second kind of their respective arguments `k`.
92
 
93
+ *Returns:* E(k), where E is given by and k is `k`.
 
 
94
 
95
  See also [[sf.cmath.ellint.2]].
96
 
97
  #### Complete elliptic integral of the third kind <a id="sf.cmath.comp.ellint.3">[[sf.cmath.comp.ellint.3]]</a>
98
 
 
103
  ```
104
 
105
  *Effects:* These functions compute the complete elliptic integral of the
106
  third kind of their respective arguments `k` and `nu`.
107
 
108
+ *Returns:* $\mathsf{\Pi}(\nu, k)$, where $\mathsf{\Pi}$ is given by , k
109
+ is `k`, and $\nu$ is `nu`.
 
110
 
111
  See also [[sf.cmath.ellint.3]].
112
 
113
  #### Regular modified cylindrical Bessel functions <a id="sf.cmath.cyl.bessel.i">[[sf.cmath.cyl.bessel.i]]</a>
114
 
 
119
  ```
120
 
121
  *Effects:* These functions compute the regular modified cylindrical
122
  Bessel functions of their respective arguments `nu` and `x`.
123
 
124
+ *Returns:* $\mathsf{I}_\nu(x)$, where $\mathsf{I}_\nu$ is given by ,
125
+ $\nu$ is `nu`, and x is `x`.
 
 
126
 
127
  *Remarks:* The effect of calling each of these functions is
128
  *implementation-defined* if `nu >= 128`.
129
 
130
  See also [[sf.cmath.cyl.bessel.j]].
 
138
  ```
139
 
140
  *Effects:* These functions compute the cylindrical Bessel functions of
141
  the first kind of their respective arguments `nu` and `x`.
142
 
143
+ *Returns:* $\mathsf{J}_\nu(x)$, where $\mathsf{J}_\nu$ is given by ,
144
+ $\nu$ is `nu`, and x is `x`.
 
145
 
146
  *Remarks:* The effect of calling each of these functions is
147
  *implementation-defined* if `nu >= 128`.
148
 
149
  #### Irregular modified cylindrical Bessel functions <a id="sf.cmath.cyl.bessel.k">[[sf.cmath.cyl.bessel.k]]</a>
 
155
  ```
156
 
157
  *Effects:* These functions compute the irregular modified cylindrical
158
  Bessel functions of their respective arguments `nu` and `x`.
159
 
160
+ *Returns:* $\mathsf{K}_\nu(x)$, where $\mathsf{K}_\nu$ is given by ,
161
+ $\nu$ is `nu`, and x is `x`.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
162
 
163
  *Remarks:* The effect of calling each of these functions is
164
  *implementation-defined* if `nu >= 128`.
165
 
166
  See also [[sf.cmath.cyl.bessel.i]], [[sf.cmath.cyl.bessel.j]],
 
176
 
177
  *Effects:* These functions compute the cylindrical Neumann functions,
178
  also known as the cylindrical Bessel functions of the second kind, of
179
  their respective arguments `nu` and `x`.
180
 
181
+ *Returns:* $\mathsf{N}_\nu(x)$, where $\mathsf{N}_\nu$ is given by ,
182
+ $\nu$ is `nu`, and x is `x`.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
183
 
184
  *Remarks:* The effect of calling each of these functions is
185
  *implementation-defined* if `nu >= 128`.
186
 
187
  See also [[sf.cmath.cyl.bessel.j]].
 
196
 
197
  *Effects:* These functions compute the incomplete elliptic integral of
198
  the first kind of their respective arguments `k` and `phi` (`phi`
199
  measured in radians).
200
 
201
+ *Returns:* F(k, φ), where F is given by , k is `k`, and φ is `phi`.
 
 
202
 
203
  #### Incomplete elliptic integral of the second kind <a id="sf.cmath.ellint.2">[[sf.cmath.ellint.2]]</a>
204
 
205
  ``` cpp
206
  floating-point-type ellint_2(floating-point-type k, floating-point-type phi);
 
210
 
211
  *Effects:* These functions compute the incomplete elliptic integral of
212
  the second kind of their respective arguments `k` and `phi` (`phi`
213
  measured in radians).
214
 
215
+ *Returns:* E(k, φ), where E is given by , k is `k`, and φ is `phi`.
 
 
216
 
217
  #### Incomplete elliptic integral of the third kind <a id="sf.cmath.ellint.3">[[sf.cmath.ellint.3]]</a>
218
 
219
  ``` cpp
220
  floating-point-type ellint_3(floating-point-type k, floating-point-type nu,
 
225
 
226
  *Effects:* These functions compute the incomplete elliptic integral of
227
  the third kind of their respective arguments `k`, `nu`, and `phi` (`phi`
228
  measured in radians).
229
 
230
+ *Returns:* $\mathsf{\Pi}(\nu, k, \phi)$, where $\mathsf{\Pi}$ is given
231
+ by , $\nu$ is `nu`, k is `k`, and φ is `phi`.
 
232
 
233
  #### Exponential integral <a id="sf.cmath.expint">[[sf.cmath.expint]]</a>
234
 
235
  ``` cpp
236
  floating-point-type expint(floating-point-type x);
 
239
  ```
240
 
241
  *Effects:* These functions compute the exponential integral of their
242
  respective arguments `x`.
243
 
244
+ *Returns:* Ei(x), where Ei is given by and x is `x`.
 
 
 
 
245
 
246
  #### Hermite polynomials <a id="sf.cmath.hermite">[[sf.cmath.hermite]]</a>
247
 
248
  ``` cpp
249
  floating-point-type hermite(unsigned n, floating-point-type x);
 
252
  ```
253
 
254
  *Effects:* These functions compute the Hermite polynomials of their
255
  respective arguments `n` and `x`.
256
 
257
+ *Returns:* Hₙ(x), where Hₙ is given by , n is `n`, and x is `x`.
 
 
 
 
258
 
259
  *Remarks:* The effect of calling each of these functions is
260
  *implementation-defined* if `n >= 128`.
261
 
262
  #### Laguerre polynomials <a id="sf.cmath.laguerre">[[sf.cmath.laguerre]]</a>
 
268
  ```
269
 
270
  *Effects:* These functions compute the Laguerre polynomials of their
271
  respective arguments `n` and `x`.
272
 
273
+ *Returns:* Lₙ(x), where Lₙ is given by , n is `n`, and x is `x`.
 
 
274
 
275
  *Remarks:* The effect of calling each of these functions is
276
  *implementation-defined* if `n >= 128`.
277
 
278
  #### Legendre polynomials <a id="sf.cmath.legendre">[[sf.cmath.legendre]]</a>
 
284
  ```
285
 
286
  *Effects:* These functions compute the Legendre polynomials of their
287
  respective arguments `l` and `x`.
288
 
289
+ *Returns:* P_ℓ(x), where P_ℓ is given by , l is `l`, and x is `x`.
 
 
 
290
 
291
  *Remarks:* The effect of calling each of these functions is
292
  *implementation-defined* if `l >= 128`.
293
 
294
  #### Riemann zeta function <a id="sf.cmath.riemann.zeta">[[sf.cmath.riemann.zeta]]</a>
 
300
  ```
301
 
302
  *Effects:* These functions compute the Riemann zeta function of their
303
  respective arguments `x`.
304
 
305
+ *Returns:* ζ(x), where ζ is given by and x is `x`.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
306
 
307
  #### Spherical Bessel functions of the first kind <a id="sf.cmath.sph.bessel">[[sf.cmath.sph.bessel]]</a>
308
 
309
  ``` cpp
310
  floating-point-type sph_bessel(unsigned n, floating-point-type x);
 
313
  ```
314
 
315
  *Effects:* These functions compute the spherical Bessel functions of the
316
  first kind of their respective arguments `n` and `x`.
317
 
318
+ *Returns:* jₙ(x), where jₙ is given by , n is `n`, and x is `x`.
 
 
319
 
320
  *Remarks:* The effect of calling each of these functions is
321
  *implementation-defined* if `n >= 128`.
322
 
323
  See also [[sf.cmath.cyl.bessel.j]].
 
332
 
333
  *Effects:* These functions compute the spherical associated Legendre
334
  functions of their respective arguments `l`, `m`, and `theta` (`theta`
335
  measured in radians).
336
 
337
+ *Returns:* Y_ℓ^m(θ, 0), where Y_ℓ^m is given by , l is `l`, m is `m`,
338
+ and θ is `theta`.
 
 
 
 
339
 
340
  *Remarks:* The effect of calling each of these functions is
341
  *implementation-defined* if `l >= 128`.
342
 
343
  See also [[sf.cmath.assoc.legendre]].
 
352
 
353
  *Effects:* These functions compute the spherical Neumann functions, also
354
  known as the spherical Bessel functions of the second kind, of their
355
  respective arguments `n` and `x`.
356
 
357
+ *Returns:* nₙ(x), where nₙ is given by , n is `n`, and x is `x`.
 
 
358
 
359
  *Remarks:* The effect of calling each of these functions is
360
  *implementation-defined* if `n >= 128`.
361
 
362
  See also [[sf.cmath.cyl.neumann]].