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This discussion follows "Adding Chebyshev polynomials to special math
functions ?". In fact the original idea of my topic has several new
elements since I get a copy of the ISO/IEC 80000-2:2009 standard (the
update of ISO 31-11)
If we look to the "Special functions" chapter, the following functions are
NOT strictly in the current C++ standard (here x is a real / a,b, c, z, w,
v are complex number / k, l, m, n are natural numbers) :
---------------------------------------------------------------------------

   - li(x) : Logarithmic integral
   (http://en.wikipedia.org/wiki/Logarithmic_integral_function)
   - Si(z) : Sine integral
   (http://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral)
   - si(x) : Complentary sine integral
   (http://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral)
   - S(z) : Fresnel integrals (with a pi/2 factor in the integrand, which
   is not the definition of wikipedia :
   http://en.wikipedia.org/wiki/Fresnel_integral but the definition from
   mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)
   - C(z) : Fresnel integrals (with a pi/2 factor in the integrand, which
   is not the definition of wikipedia :
   http://en.wikipedia.org/wiki/Fresnel_integral but the definition from
   mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)
   - F(a, b, c, z) : Hypergeometric functions
   (http://en.wikipedia.org/wiki/Hypergeometric_function)
   - F(a, c, z) : Confluent hypergeometric function
   (http://en.wikipedia.org/wiki/Confluent_hypergeometric_function)
   - Y(l, m, theta, phi) : Spherical harmonics
   (http://en.wikipedia.org/wiki/Spherical_harmonics)
   - T(n, z) : Chebyshev polynomial of the first kind
   (http://en.wikipedia.org/wiki/Chebyshev_polynomials)
   - U(n, z) : Chebyshev polynomial of the second kind
   (http://en.wikipedia.org/wiki/Chebyshev_polynomials)
   - H1(v, z) : Hankel functions
   (http://en.wikipedia.org/wiki/Hankel_function#Hankel_functions:_H.CE.B1)
   - H2(v, z) : Hankel functions
   (http://en.wikipedia.org/wiki/Hankel_function#Hankel_functions:_H.CE.B1)
   - h1(l, z) : Spherical Hankel functions
   (http://en.wikipedia.org/wiki/Bessel_function#Spherical_Hankel_functions:_hn)
   - h2(l, z) : Spherical Hankel functions
   (http://en.wikipedia.org/wiki/Bessel_function#Spherical_Hankel_functions:_hn)
   - Ai(z) : Airy functions (http://en.wikipedia.org/wiki/Airy_function)
   - Bi(z) : Airy functions (http://en.wikipedia.org/wiki/Airy_function)

---------------------------------------------------------------------------

Now, if we exclude strictly complex functions, the function that remain and
that are not included in the current standard are :

---------------------------------------------------------------------------

   - li(x) : Logarithmic integral
   (http://en.wikipedia.org/wiki/Logarithmic_integral_function)
   - Si(x) : Sine integral
   (http://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral)
   - si(x) : Complentary sine integral
   (http://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral)
   - S(x) : Fresnel integrals (with a pi/2 factor in the integrand, which
   is not the definition of wikipedia :
   http://en.wikipedia.org/wiki/Fresnel_integral but the definition from
   mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)
   - C(x) : Fresnel integrals (with a pi/2 factor in the integrand, which
   is not the definition of wikipedia :
   http://en.wikipedia.org/wiki/Fresnel_integral but the definition from
   mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)
   - F(a, b, c, x) : Hypergeometric functions
   (http://en.wikipedia.org/wiki/Hypergeometric_function)
   - F(a, c, x) : Confluent hypergeometric function
   (http://en.wikipedia.org/wiki/Confluent_hypergeometric_function)
   - T(n, x) : Chebyshev polynomial of the first kind
   (http://en.wikipedia.org/wiki/Chebyshev_polynomials)
   - U(n, x) : Chebyshev polynomial of the second kind
   (http://en.wikipedia.org/wiki/Chebyshev_polynomials)
   - Ai(x) : Airy functions (http://en.wikipedia.org/wiki/Airy_function)
   - Bi(x) : Airy functions (http://en.wikipedia.org/wiki/Airy_function)

---------------------------------------------------------------------------

So, it would be great to include these functions and maybe some related
functions.
In the preceding list, the inclusion of sine and complementary sine
integral, can be a problem because there are some other trigonometric
integral functions that may be included as a complement : Ci, ci, Cin, Chi,
Shi (see (http://en.wikipedia.org/wiki/Trigonometric_integral) )?
But all the other are well defined.

As a complement of these functions, some related functions could be
included, even if they does not appear in the ISO 80000 document :

---------------------------------------------------------------------------

   - sinc(x) : Cardinal sine (http://en.wikipedia.org/wiki/Sinc_function)
   as a function widely used in optics and as a function related to Si(x) in
   the preceding list
   - V(n, x) : Chebyshev polynomial of the third kind because there are 4
   kinds of Chebyshev polynomials and not only two as in the preceding list
   - W(n, x) : Chebyshev polynomial of the fourth kind because there are 4
   kinds of Chebyshev polynomials and not only two as in the preceding list
   - P(alpha, beta, n, x) : Jacobi polynomial as a generalization of Gegenbauer
   , Chebyshev, Legendre and Zernike polynomials
   (http://en.wikipedia.org/wiki/Jacobi_polynomials)
   - Z(n, m, rho, phi) : Zernike polynomials as a complement of Chebyshev
   and Legendre polynomials (http://en.wikipedia.org/wiki/Zernike_polynomials)
   - C(alpha, n, x) : Gegenbauer polynomials as a generalization of Chebyshev
   and Legendre polynomials(http://en.wikipedia.org/wiki/Gegenbauer_polynomials)

---------------------------------------------------------------------------

The key point is that these functions (except the last list) are now
standardized by an ISO document, so I think that it would facilitate their
recognition. I open the topic to get some advices/reflexions/critics before
writing a proposal.

Thank you very much.

--




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This discussion follows "<span id=3D"t-t" class=3D"GPW32EMDDMB">Adding Cheb=
yshev polynomials to special math functions ?". In fact the original idea o=
f my topic has several new elements since I get a copy of the ISO/IEC 80000=
-2:2009 standard (the update of ISO 31-11)<br>If we look to the "Special fu=
nctions" chapter, the following functions are NOT strictly in the current C=
++ standard (here x is a real / a,b, c, z, w, v are complex number / k, l, =
m, n are natural numbers) :<br>--------------------------------------------=
-------------------------------<br></span><ul><li><span id=3D"t-t" class=3D=
"GPW32EMDDMB">li(x) : Logarithmic integral (http://en.wikipedia.org/wiki/Lo=
garithmic_integral_function)</span></li><li><span id=3D"t-t" class=3D"GPW32=
EMDDMB">Si(z) : Sine integral (http://en.wikipedia.org/wiki/Trigonometric_i=
ntegral#Sine_integral)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB=
">si(x) : Complentary sine integral </span><span id=3D"t-t" class=3D"GPW32E=
MDDMB">(http://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral)<=
/span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">S(z) :</span><span id=
=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"> Fres=
nel integrals (with a pi/2 factor
 in the integrand, which is not the definition of wikipedia :=20
http://en.wikipedia.org/wiki/Fresnel_integral but the definition from=20
mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)</span></spa=
n></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">C(z) : Fresnel integrals =
(with a pi/2 factor in the integrand, which is not the definition of wikipe=
dia : http://en.wikipedia.org/wiki/Fresnel_integral but the definition from=
 mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)</span></li=
><li><span id=3D"t-t" class=3D"GPW32EMDDMB">F(a, b, c, z) : Hypergeometric =
functions (http://en.wikipedia.org/wiki/Hypergeometric_function)</span></li=
><li><span id=3D"t-t" class=3D"GPW32EMDDMB">F(a, c, z) : Confluent hypergeo=
metric function (http://en.wikipedia.org/wiki/Confluent_hypergeometric_func=
tion)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">Y(l, m, theta, =
phi) : Spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonic=
s)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">T(n, z) : Chebyshe=
v polynomial of the first kind (http://en.wikipedia.org/wiki/Chebyshev_poly=
nomials)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">U(n, z) : Ch=
ebyshev polynomial of the second kind (http://en.wikipedia.org/wiki/Chebysh=
ev_polynomials)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">H1(v,=
 z) : Hankel functions (http://en.wikipedia.org/wiki/Hankel_function#Hankel=
_functions:_H.CE.B1)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">=
<span id=3D"t-t" class=3D"GPW32EMDDMB">H2(v, z) : Hankel functions (http://=
en.wikipedia.org/wiki/Hankel_function#Hankel_functions:_H.CE.B1)</span></sp=
an></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">h1(l, z) : Spherical Han=
kel functions (http://en.wikipedia.org/wiki/Bessel_function#Spherical_Hanke=
l_functions:_hn)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB"><spa=
n id=3D"t-t" class=3D"GPW32EMDDMB">h2(l, z) :</span></span><span id=3D"t-t"=
 class=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t=
-t" class=3D"GPW32EMDDMB"> Spherical </span> Hankel functions (http://en.wi=
kipedia.org/wiki/Bessel_function#Spherical_Hankel_functions:_hn)</span></sp=
an></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" class=
=3D"GPW32EMDDMB">Ai(z) : Airy functions (http://en.wikipedia.org/wiki/Airy_=
function)</span></span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB"><spa=
n id=3D"t-t" class=3D"GPW32EMDDMB">Bi(z) : Airy functions (http://en.wikipe=
dia.org/wiki/Airy_function)</span></span></li></ul><span id=3D"t-t" class=
=3D"GPW32EMDDMB">----------------------------------------------------------=
-----------------</span><br><br>Now, if we exclude strictly complex functio=
ns, the function that remain and that are not included in the current stand=
ard are :<br><br><span id=3D"t-t" class=3D"GPW32EMDDMB">-------------------=
--------------------------------------------------------<br></span><ul><li>=
<span id=3D"t-t" class=3D"GPW32EMDDMB">li(x) : Logarithmic integral (http:/=
/en.wikipedia.org/wiki/Logarithmic_integral_function)</span></li><li><span =
id=3D"t-t" class=3D"GPW32EMDDMB">Si(x) : Sine integral (http://en.wikipedia=
..org/wiki/Trigonometric_integral#Sine_integral)</span></li><li><span id=3D"=
t-t" class=3D"GPW32EMDDMB">si(x) : Complentary sine integral </span><span i=
d=3D"t-t" class=3D"GPW32EMDDMB">(http://en.wikipedia.org/wiki/Trigonometric=
_integral#Sine_integral)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDD=
MB">S(x) :</span><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" cl=
ass=3D"GPW32EMDDMB"> Fresnel integrals (with a pi/2 factor
 in the integrand, which is not the definition of wikipedia :=20
http://en.wikipedia.org/wiki/Fresnel_integral but the definition from=20
mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)</span></spa=
n></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">C(x)
 : Fresnel integrals (with a pi/2 factor in the integrand, which is not=20
the definition of wikipedia :=20
http://en.wikipedia.org/wiki/Fresnel_integral but the definition from=20
mathworld : http://mathworld.wolfram.com/FresnelIntegrals.html)</span></li>=
<li><span id=3D"t-t" class=3D"GPW32EMDDMB">F(a, b, c, x) : Hypergeometric f=
unctions (http://en.wikipedia.org/wiki/Hypergeometric_function)</span></li>=
<li><span id=3D"t-t" class=3D"GPW32EMDDMB">F(a, c, x) : Confluent hypergeom=
etric function (http://en.wikipedia.org/wiki/Confluent_hypergeometric_funct=
ion)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">T(n, x) : Chebys=
hev polynomial of the first kind (http://en.wikipedia.org/wiki/Chebyshev_po=
lynomials)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">U(n, x) : =
Chebyshev polynomial of the second kind (http://en.wikipedia.org/wiki/Cheby=
shev_polynomials)</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB"><sp=
an id=3D"t-t" class=3D"GPW32EMDDMB">Ai(x) : Airy functions (http://en.wikip=
edia.org/wiki/Airy_function)</span></span></li><li><span id=3D"t-t" class=
=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB">Bi(x) : Airy functi=
ons (http://en.wikipedia.org/wiki/Airy_function)</span></span></li></ul><sp=
an id=3D"t-t" class=3D"GPW32EMDDMB">---------------------------------------=
------------------------------------</span><br><br>So, it would be great to=
 include these functions and maybe some related functions. <br>In the prece=
ding list, the inclusion of sine and complementary sine integral, can be a =
problem because there are some other trigonometric integral functions that =
may be included as a complement : Ci, ci, Cin, Chi, Shi (see <span id=3D"t-=
t" class=3D"GPW32EMDDMB">(http://en.wikipedia.org/wiki/Trigonometric_integr=
al) </span>)?<br>But all the other are well defined.<br><br>As a complement=
 of these functions, some related functions could be included, even if they=
 does not appear in the ISO 80000 document :<br><br><span id=3D"t-t" class=
=3D"GPW32EMDDMB">----------------------------------------------------------=
-----------------<br></span><ul><li><span id=3D"t-t" class=3D"GPW32EMDDMB">=
sinc(x) : Cardinal sine (http://en.wikipedia.org/wiki/Sinc_function) as a f=
unction widely used in optics and as a function related to Si(x) in the pre=
ceding list<br></span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB">V(n, =
x) : Chebyshev polynomial of the third kind because there are 4 kinds of Ch=
ebyshev polynomials and not only two as in the preceding list<br></span></l=
i><li><span id=3D"t-t" class=3D"GPW32EMDDMB">W(n, x) : Chebyshev polynomial=
 of the fourth kind </span><span id=3D"t-t" class=3D"GPW32EMDDMB">because t=
here are 4 kinds of Chebyshev polynomials and not only two </span>as in the=
 preceding list<br></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=
=3D"t-t" class=3D"GPW32EMDDMB">P(alpha, beta, n, x) : Jacobi polynomial as =
a generalization of </span></span><span id=3D"t-t" class=3D"GPW32EMDDMB"><s=
pan id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"=
><span id=3D"t-t" class=3D"GPW32EMDDMB"></span></span><span id=3D"t-t" clas=
s=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" c=
lass=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB">Gegenbauer , </=
span></span></span></span>Chebyshev, Legendre and Zernike polynomials (http=
://en.wikipedia.org/wiki/Jacobi_polynomials)<br></span></span></li><li><spa=
n id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB">Z=
(n, m, rho, phi) : Zernike polynomials as a complement of Chebyshev and Leg=
endre polynomials (http://en.wikipedia.org/wiki/Zernike_polynomials)</span>=
</span></li><li><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" cla=
ss=3D"GPW32EMDDMB">C(alpha, n, x) : </span></span><span id=3D"t-t" class=3D=
"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" class=
=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB">Gegenbauer polynomi=
als as a generalization of </span></span></span></span><span id=3D"t-t" cla=
ss=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-t" =
class=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB"><span id=3D"t-=
t" class=3D"GPW32EMDDMB"><span id=3D"t-t" class=3D"GPW32EMDDMB">Chebyshev a=
nd Legendre polynomials</span></span></span></span> (http://en.wikipedia.or=
g/wiki/Gegenbauer_polynomials)<br></span></span></li></ul><span id=3D"t-t" =
class=3D"GPW32EMDDMB">-----------------------------------------------------=
----------------------</span><br><br>The key point is that these functions =
(except the last list) are now standardized by an ISO document, so I think =
that it would facilitate their recognition. I open the topic to get some ad=
vices/reflexions/critics before writing a proposal.<br><br>Thank you very m=
uch.<br><br>

<p></p>

-- <br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />

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.