tmp/tmpwq_9aap0/{from.md → to.md}
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#### Spherical Neumann functions <a id="sf.cmath.sph_neumann">[[sf.cmath.sph_neumann]]</a>
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``` cpp
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double sph_neumann(unsigned n, double x);
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float sph_neumannf(unsigned n, float x);
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long double sph_neumannl(unsigned n, long double x);
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```
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*Effects:* These functions compute the spherical Neumann functions, also
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known as the spherical Bessel functions of the second kind, of their
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respective arguments `n` and `x`.
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*Returns:* $$%
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\mathsf{n}_n(x) =
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(\pi/2x)^{1\!/\!2} \mathsf{N}_{n + 1\!/\!2}(x),
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\quad \mbox{for $x \ge 0$}$$ where n is `n` and x is `x`.
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*Remarks:* The effect of calling each of these functions is
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*implementation-defined* if `n >= 128`.
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See also [[sf.cmath.cyl_neumann]].
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<!-- Link reference definitions -->
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[accumulate]: #accumulate
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[adjacent.difference]: #adjacent.difference
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[algorithms]: algorithms.md#algorithms
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[bad.alloc]: language.md#bad.alloc
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| 28 |
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[basic.fundamental]: basic.md#basic.fundamental
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| 29 |
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[basic.stc.thread]: basic.md#basic.stc.thread
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[basic.types]: basic.md#basic.types
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[c.math]: #c.math
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| 32 |
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[c.math.abs]: #c.math.abs
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[c.math.fpclass]: #c.math.fpclass
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[c.math.hypot3]: #c.math.hypot3
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[c.math.rand]: #c.math.rand
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| 36 |
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[cfenv]: #cfenv
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[cfenv.syn]: #cfenv.syn
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| 38 |
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[class.gslice]: #class.gslice
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[class.gslice.overview]: #class.gslice.overview
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[class.slice]: #class.slice
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[class.slice.overview]: #class.slice.overview
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| 42 |
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[cmath.syn]: #cmath.syn
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| 43 |
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[cmplx.over]: #cmplx.over
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| 44 |
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[complex]: #complex
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[complex.literals]: #complex.literals
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[complex.member.ops]: #complex.member.ops
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[complex.members]: #complex.members
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| 48 |
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[complex.numbers]: #complex.numbers
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| 49 |
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[complex.ops]: #complex.ops
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| 50 |
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[complex.special]: #complex.special
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[complex.syn]: #complex.syn
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[complex.transcendentals]: #complex.transcendentals
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| 53 |
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[complex.value.ops]: #complex.value.ops
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| 54 |
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[cons.slice]: #cons.slice
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[conv.prom]: conv.md#conv.prom
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| 56 |
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[cpp.pragma]: cpp.md#cpp.pragma
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| 57 |
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[cstdlib.syn]: language.md#cstdlib.syn
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| 58 |
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[dcl.array]: dcl.md#dcl.array
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| 59 |
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[dcl.init]: dcl.md#dcl.init
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[exclusive.scan]: #exclusive.scan
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[function.objects]: utilities.md#function.objects
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[gslice.access]: #gslice.access
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[gslice.array.assign]: #gslice.array.assign
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| 64 |
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[gslice.array.comp.assign]: #gslice.array.comp.assign
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| 65 |
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[gslice.array.fill]: #gslice.array.fill
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| 66 |
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[gslice.cons]: #gslice.cons
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[implimits]: limits.md#implimits
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[inclusive.scan]: #inclusive.scan
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[indirect.array.assign]: #indirect.array.assign
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[indirect.array.comp.assign]: #indirect.array.comp.assign
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[indirect.array.fill]: #indirect.array.fill
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| 72 |
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[inner.product]: #inner.product
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[input.iterators]: iterators.md#input.iterators
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[input.output]: input.md#input.output
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[iostate.flags]: input.md#iostate.flags
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[istream.formatted]: input.md#istream.formatted
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[iterator.requirements.general]: iterators.md#iterator.requirements.general
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[library.c]: library.md#library.c
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[mask.array.assign]: #mask.array.assign
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[mask.array.comp.assign]: #mask.array.comp.assign
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[mask.array.fill]: #mask.array.fill
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[numarray]: #numarray
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| 83 |
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[numeric.iota]: #numeric.iota
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[numeric.ops]: #numeric.ops
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[numeric.ops.gcd]: #numeric.ops.gcd
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[numeric.ops.lcm]: #numeric.ops.lcm
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| 87 |
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[numeric.ops.overview]: #numeric.ops.overview
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[numeric.requirements]: #numeric.requirements
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[numerics]: #numerics
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[numerics.defns]: #numerics.defns
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[numerics.general]: #numerics.general
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| 92 |
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[output.iterators]: iterators.md#output.iterators
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[partial.sum]: #partial.sum
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[rand]: #rand
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[rand.adapt]: #rand.adapt
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[rand.adapt.disc]: #rand.adapt.disc
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[rand.adapt.general]: #rand.adapt.general
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[rand.adapt.ibits]: #rand.adapt.ibits
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[rand.adapt.shuf]: #rand.adapt.shuf
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[rand.device]: #rand.device
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[rand.dist]: #rand.dist
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[rand.dist.bern]: #rand.dist.bern
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[rand.dist.bern.bernoulli]: #rand.dist.bern.bernoulli
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[rand.dist.bern.bin]: #rand.dist.bern.bin
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[rand.dist.bern.geo]: #rand.dist.bern.geo
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[rand.dist.bern.negbin]: #rand.dist.bern.negbin
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[rand.dist.general]: #rand.dist.general
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[rand.dist.norm]: #rand.dist.norm
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[rand.dist.norm.cauchy]: #rand.dist.norm.cauchy
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[rand.dist.norm.chisq]: #rand.dist.norm.chisq
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[rand.dist.norm.f]: #rand.dist.norm.f
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[rand.dist.norm.lognormal]: #rand.dist.norm.lognormal
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[rand.dist.norm.normal]: #rand.dist.norm.normal
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[rand.dist.norm.t]: #rand.dist.norm.t
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[rand.dist.pois]: #rand.dist.pois
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[rand.dist.pois.exp]: #rand.dist.pois.exp
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[rand.dist.pois.extreme]: #rand.dist.pois.extreme
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[rand.dist.pois.gamma]: #rand.dist.pois.gamma
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[rand.dist.pois.poisson]: #rand.dist.pois.poisson
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[rand.dist.pois.weibull]: #rand.dist.pois.weibull
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[rand.dist.samp]: #rand.dist.samp
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[rand.dist.samp.discrete]: #rand.dist.samp.discrete
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[rand.dist.samp.pconst]: #rand.dist.samp.pconst
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[rand.dist.samp.plinear]: #rand.dist.samp.plinear
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[rand.dist.uni]: #rand.dist.uni
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| 126 |
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[rand.dist.uni.int]: #rand.dist.uni.int
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[rand.dist.uni.real]: #rand.dist.uni.real
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[rand.eng]: #rand.eng
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[rand.eng.lcong]: #rand.eng.lcong
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[rand.eng.mers]: #rand.eng.mers
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[rand.eng.sub]: #rand.eng.sub
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[rand.predef]: #rand.predef
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[rand.req]: #rand.req
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[rand.req.adapt]: #rand.req.adapt
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| 135 |
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[rand.req.dist]: #rand.req.dist
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| 136 |
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[rand.req.eng]: #rand.req.eng
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[rand.req.genl]: #rand.req.genl
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[rand.req.seedseq]: #rand.req.seedseq
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[rand.req.urng]: #rand.req.urng
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[rand.synopsis]: #rand.synopsis
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[rand.util]: #rand.util
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[rand.util.canonical]: #rand.util.canonical
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[rand.util.seedseq]: #rand.util.seedseq
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[random.access.iterators]: iterators.md#random.access.iterators
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[reduce]: #reduce
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[res.on.data.races]: library.md#res.on.data.races
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[sf.cmath]: #sf.cmath
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| 148 |
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[sf.cmath.assoc_laguerre]: #sf.cmath.assoc_laguerre
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| 149 |
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[sf.cmath.assoc_legendre]: #sf.cmath.assoc_legendre
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| 150 |
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[sf.cmath.beta]: #sf.cmath.beta
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[sf.cmath.comp_ellint_1]: #sf.cmath.comp_ellint_1
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[sf.cmath.comp_ellint_2]: #sf.cmath.comp_ellint_2
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| 153 |
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[sf.cmath.comp_ellint_3]: #sf.cmath.comp_ellint_3
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| 154 |
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[sf.cmath.cyl_bessel_i]: #sf.cmath.cyl_bessel_i
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[sf.cmath.cyl_bessel_j]: #sf.cmath.cyl_bessel_j
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| 156 |
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[sf.cmath.cyl_bessel_k]: #sf.cmath.cyl_bessel_k
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| 157 |
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[sf.cmath.cyl_neumann]: #sf.cmath.cyl_neumann
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[sf.cmath.ellint_1]: #sf.cmath.ellint_1
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[sf.cmath.ellint_2]: #sf.cmath.ellint_2
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[sf.cmath.ellint_3]: #sf.cmath.ellint_3
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[sf.cmath.expint]: #sf.cmath.expint
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[sf.cmath.hermite]: #sf.cmath.hermite
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[sf.cmath.laguerre]: #sf.cmath.laguerre
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| 164 |
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[sf.cmath.legendre]: #sf.cmath.legendre
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[sf.cmath.riemann_zeta]: #sf.cmath.riemann_zeta
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[sf.cmath.sph_bessel]: #sf.cmath.sph_bessel
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| 167 |
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[sf.cmath.sph_legendre]: #sf.cmath.sph_legendre
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[sf.cmath.sph_neumann]: #sf.cmath.sph_neumann
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[slice.access]: #slice.access
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[slice.arr.assign]: #slice.arr.assign
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[slice.arr.comp.assign]: #slice.arr.comp.assign
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[slice.arr.fill]: #slice.arr.fill
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[strings]: strings.md#strings
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[tab:RandomDistribution]: #tab:RandomDistribution
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[tab:RandomEngine]: #tab:RandomEngine
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| 176 |
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[tab:SeedSequence]: #tab:SeedSequence
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[tab:UniformRandomBitGenerator]: #tab:UniformRandomBitGenerator
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[tab:copyassignable]: #tab:copyassignable
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[tab:copyconstructible]: #tab:copyconstructible
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[tab:equalitycomparable]: #tab:equalitycomparable
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[tab:iterator.input.requirements]: iterators.md#tab:iterator.input.requirements
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[tab:moveassignable]: #tab:moveassignable
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[tab:moveconstructible]: #tab:moveconstructible
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[tab:numerics.lib.summary]: #tab:numerics.lib.summary
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[template.gslice.array]: #template.gslice.array
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[template.gslice.array.overview]: #template.gslice.array.overview
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[template.indirect.array]: #template.indirect.array
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[template.indirect.array.overview]: #template.indirect.array.overview
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[template.mask.array]: #template.mask.array
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[template.mask.array.overview]: #template.mask.array.overview
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[template.slice.array]: #template.slice.array
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[template.slice.array.overview]: #template.slice.array.overview
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[template.valarray]: #template.valarray
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[template.valarray.overview]: #template.valarray.overview
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[thread.thread.class]: thread.md#thread.thread.class
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[transform.exclusive.scan]: #transform.exclusive.scan
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[transform.inclusive.scan]: #transform.inclusive.scan
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[transform.reduce]: #transform.reduce
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[valarray.access]: #valarray.access
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[valarray.assign]: #valarray.assign
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[valarray.binary]: #valarray.binary
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[valarray.cassign]: #valarray.cassign
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[valarray.comparison]: #valarray.comparison
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[valarray.cons]: #valarray.cons
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[valarray.members]: #valarray.members
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[valarray.nonmembers]: #valarray.nonmembers
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[valarray.range]: #valarray.range
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[valarray.special]: #valarray.special
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[valarray.sub]: #valarray.sub
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[valarray.syn]: #valarray.syn
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[valarray.transcend]: #valarray.transcend
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[valarray.unary]: #valarray.unary
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[vector]: containers.md#vector
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[^1]: In other words, value types. These include arithmetic types,
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pointers, the library class `complex`, and instantiations of
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`valarray` for value types.
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[^2]: The name of this engine refers, in part, to a property of its
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period: For properly-selected values of the parameters, the period
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is closely related to a large Mersenne prime number.
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[^3]: The parameter is intended to allow an implementation to
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differentiate between different sources of randomness.
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[^4]: If a device has n states whose respective probabilities are
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P₀, …, Pₙ₋₁, the device entropy S is defined as
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$S = - \sum_{i=0}^{n-1} P_i \cdot \log P_i$.
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[^5]: b is introduced to avoid any attempt to produce more bits of
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randomness than can be held in `RealType`.
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[^6]: The distribution corresponding to this probability density
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function is also known (with a possible change of variable) as the
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Gumbel Type I, the log-Weibull, or the Fisher-Tippett Type I
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distribution.
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[^7]: Annex [[implimits]] recommends a minimum number of recursively
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nested template instantiations. This requirement thus indirectly
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suggests a minimum allowable complexity for valarray expressions.
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[^8]: The intent is to specify an array template that has the minimum
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functionality necessary to address aliasing ambiguities and the
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proliferation of temporaries. Thus, the `valarray` template is
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neither a matrix class nor a field class. However, it is a very
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useful building block for designing such classes.
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[^9]: This default constructor is essential, since arrays of `valarray`
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may be useful. After initialization, the length of an empty array
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can be increased with the `resize` member function.
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[^10]: This constructor is the preferred method for converting a C array
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to a `valarray` object.
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[^11]: This copy constructor creates a distinct array rather than an
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alias. Implementations in which arrays share storage are permitted,
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but they shall implement a copy-on-reference mechanism to ensure
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that arrays are conceptually distinct.
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[^12]: BLAS stands for *Basic Linear Algebra Subprograms.* C++programs
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may instantiate this class. See, for example, Dongarra, Du Croz,
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Duff, and Hammerling: *A set of Level 3 Basic Linear Algebra
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Subprograms*; Technical Report MCS-P1-0888, Argonne National
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Laboratory (USA), Mathematics and Computer Science Division, August,
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1988.
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[^13]: The use of fully closed ranges is intentional.
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[^14]: `accumulate` is similar to the APL reduction operator and Common
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Lisp reduce function, but it avoids the difficulty of defining the
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result of reduction on an empty sequence by always requiring an
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initial value.
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[^15]: The use of fully closed ranges is intentional.
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[^16]: The use of fully closed ranges is intentional.
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[^17]: The use of fully closed ranges is intentional.
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[^18]: A mathematical function is mathematically defined for a given set
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of argument values (a) if it is explicitly defined for that set of
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argument values, or (b) if its limiting value exists and does not
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depend on the direction of approach.
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