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Howard Cohl - math NIST - National Institute of Standards and
Transcript of Howard Cohl - math NIST - National Institute of Standards and
parameter di�erentiation for Bessel andassociated Legendre functions
Howard S. Cohl?†
? Information Technology Laboratory, National Institute of Standards and Technology,
Gaithersburg, Maryland, U.S.A.
† Department of Mathematics, University of Auckland, Auckland, New Zealand
International Conference on Special Functionsin the 21st Century: Theory and Applications
Washington, D.C., U.S.A.
April 7, 2011
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 1 / 31
Outline
1 introduction � special functions
2 parameter di�erentiation � Bessel functions
3 parameter di�erentiation � associated Legendre functions
4 some particular cases for Legendre functions
5 references
6 summary
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 2 / 31
introduction � special functions
de�nitions: special functions
gamma function (Re z > 0) Γ(z + 1) = zΓ(z)
Γ(z) :=∫ ∞
0tz−1e−tdt
Euler re�ection formula: Γ(z)Γ(1− z) =π
sinπzfactorial: (for n ∈ {0, 1, 2, . . .}) Γ(n+ 1) = n!
Pochhammer symbol: (for n ∈ {0, 1, 2, . . .}) (a)n :=Γ(a+ n)
Γ(a)digamma function � logarithmic derivative of gamma function
d
dzΓ(z) =: Γ(z)ψ(z)
generalized factorial function
pFq ((ap); (bq); z) =∞∑n=0
(a1)n(a2)n . . . (ap)n(b1)n(b2)n . . . (bq)n
zk
k!
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 3 / 31
introduction � special functions
de�nitions � Bessel functions (unrestricted order ν)
Bessel function of the �rst kind
Jν(z) =(z/2)ν
Γ(ν + 1)0F1
(ν + 1;−z
2
4
)=(z
2
)ν ∞∑n=0
(−z2/4)n
n!Γ(ν + n+ 1)
Bessel function of the second kind (Weber's function)
Yν(z) = cot(πν)Jν(z)− csc(πν)J−ν(z)Yn(z) = 1
π∂Jν(z)∂ν
∣∣∣ν=n
+ (−1)n
π∂Jν(z)∂ν
∣∣∣ν=−n
modi�ed Bessel function of the �rst kind
Iν(z) =(z/2)ν
Γ(ν + 1)0F1
(ν + 1;
z2
4
)=(z
2
)ν ∞∑n=0
(z2/4)n
n!Γ(ν + n+ 1)
modi�ed Bessel function of the second kind (Macdonald's function)
Kν(z) = π2 csc(πν)I−ν(z)− csc(πν)Iν(z)
Kn(z) = (−1)n
2∂I−ν(z)∂ν
∣∣∣ν=n− (−1)n
2∂Iν(z)∂ν
∣∣∣ν=n
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 4 / 31
introduction � special functions
associated Legendre functions P µν , Q
µν : C \ {−1, 1} → C
(complex degree ν order µ)
two (Gauss hypergeometric) functions � associated Legendre d.e.
(1− z2)d2w
dz2− 2z
dw
dz+[ν(ν + 1)− µ2
1− z2
]w = 0
associated Legendre function of the �rst kind (|1− z| < 2)
Pµν (z) = 1Γ(1−µ)
[z+1z−1
]µ2
2F1
(−ν, ν + 1; 1− µ; 1−z
2
)= − sin(πν)
π
(z+1z−1
)µ2∞∑n=0
Γ(−ν + n)Γ(ν + 1 + n)n!Γ(1− µ+ n)
(1− z
2
)nassociated Legendre function of the second kind (|z| > 1)
Qµν (z) =√πeiµπΓ(ν+µ+1)(z2−1)µ/2
2ν+1Γ(ν+ 32)zν+µ+1 2F1
(ν+µ+2
2 , ν+µ+12 ; ν + 3
2 ; 1z2
)= 1
2eiπµ(z2 − 1)
µ2
∞∑n=0
Γ(ν+µ+22 + n)Γ(ν+µ+1
2 + n)n!Γ(ν + 3
2 + n)z2n
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 5 / 31
introduction � special functions
order derivative illustration for modi�ed Bessel functions
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 6 / 31
parameter di�erentiation � Bessel functions
order derivatives for Bessel functions
Bessel function of the �rst kind
∂J±ν(z)∂ν
= ±J±ν(z) log(z
2
)∓∞∑m=0
(−1)m(z
2
)±ν+2m ψ(±ν +m+ 1)m!Γ(±ν +m+ 1)
Bessel function of the second kind (Weber's function)
∂Yν(z)∂ν
= cot(πν)∂Jν(z)∂ν
− csc(πν)∂J−ν(z)∂ν
− π csc(πν)Y−ν(z)
modi�ed Bessel function of the �rst kind
∂Iν(z)∂ν
= Iν(z) log(z
2
)−∞∑m=0
(z2
)ν+2m ψ(ν +m+ 1)m!Γ(ν +m+ 1)
modi�ed Bessel function of the second kind (Macdonald's function)
∂Kν(z)∂ν
= −π cot(πν)Kν(z) +π
2csc(πν)
[∂I−ν(z)∂ν
− ∂Iν(z)∂ν
]Howard Cohl (NIST) parameter di�erentiation April 7, 2011 7 / 31
parameter di�erentiation � Bessel functions
Bessel function mirror symmetry about ν = 0
Bessel function of the �rst kind
J−ν(z) = Jν(z) cos(πν)− Yν(z) sin(πν)
Bessel function of the second kind (Weber's function)
Y−ν(z) = Jν(z) sin(πν) + Yν(z) cos(πν)
modi�ed Bessel function of the �rst kind
I−ν(z) = Iν(z) +2πKν(z) sin(πν)
modi�ed Bessel function of the second kind (Macdonald's function)
K−ν(z) = Kν(z)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 8 / 31
parameter di�erentiation � Bessel functions
order derivatives for Bessel functions (zero-order)
Derivatives of Bessel function with respect to the order evaluated at
integer-orders is given in �3.2.3 in Magnus, Oberhettinger & Soni (1966)
Bessel function of the �rst kind[∂
∂νJν(z)
]ν=0
=π
2Y0(z)
Bessel function of the second kind (Weber's function)[∂
∂νYν(z)
]ν=0
= −π2J0(z)
modi�ed Bessel function of the �rst kind[∂
∂νIν(z)
]ν=0
= −K0(z)
modi�ed Bessel function of the second kind (Macdonald's function)[∂
∂νKν(z)
]ν=0
= 0
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 9 / 31
parameter di�erentiation � Bessel functions
order derivatives for Bessel functions (integer-order)
Derivatives of Bessel function with respect to the order evaluated at
integer-orders is given by (�3.2.3 in MOS (1966))
Bessel function of the �rst kind[∂
∂νJν(z)
]ν=±n
=π
2(±1)nYn(z)± (±1)n
n!2
n−1∑k=0
(z/2)k−n
k!(n− k)Jk(z)
Bessel function of the second kind (Weber's function)[∂
∂νYν(z)
]ν=±n
= −π2
(±1)nJn(z)± (±1)nn!2
n−1∑k=0
(z/2)k−n
k!(n− k)Yk(z)
modi�ed Bessel function of the �rst kind[∂
∂νIν(z)
]ν=±n
= (−1)n+1Kn(z)± n!2
n−1∑k=0
(−z/2)k−n
k!(n− k)Ik(z)
modi�ed Bessel function of the second kind (Macdonald's function)[∂
∂νKν(z)
]ν=±n
= ±n!2
n−1∑k=0
(z/2)k−n
k!(n− k)Kk(z)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 10 / 31
parameter di�erentiation � Bessel functions
order derivatives for Bessel functions (half-integer)
Derivatives of Bessel function with respect to the order evaluated at
half-integer-orders is given by (�3.3.3 in MOS (1966))
Bessel function of the �rst kind[∂
∂νJν(z)
]ν=±1/2
=
√2πz
[{sin zcos z
}Ci(2z)∓
{cos zsin z
}Si(2z)
]Bessel function of the second kind (Weber's function)[∂
∂νYν(z)
]ν=±1/2
= ±√
2πz
[{cos zsin z
}Ci(2z)±
{sin zcos z
}[Si(2z)− π]
]modi�ed Bessel function of the �rst kind[∂
∂νIν(z)
]ν=±1/2
=
√1
2πz[ezEi(−2z)∓ e−zEi(2z)
]modi�ed Bessel function of the second kind (Macdonald's function)[∂
∂νKν(z)
]ν=±1/2
= ∓√
π
2zezEi(−2z)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 11 / 31
parameter di�erentiation � Bessel functions
order derivatives for Bessel functions
cosine integral: Ci(z) := −∫ ∞z
cos ttdt
sine integral: Si(z) :=∫ z
0
sin ttdt
exponential integral: Ei(z) := −∫ ∞−z
e−t
tdt
order derivatives at half-odd integer orders[∂
∂νJν(z)
]ν=1/2+n
= ci(2z)Jn+1/2(z)− (−1)nSi(2z)J−n−1/2(z)
+n!2
n−1∑k=0
(z/2)k−n
k!(n− k)Jk+1/2(z)− n!
√πz
2
n∑k=1
(2/z)k
(n− k)!k
k−1∑p=0
zp
p!
×[Jn−k+1/2(z)Jp−1/2(2z)− (−1)n−k−pJk−n−1/2(z)J1/2−p(2z)
]Howard Cohl (NIST) parameter di�erentiation April 7, 2011 12 / 31
parameter di�erentiation � associated Legendre functions
parameter di�erentiation for associated Legendre functions
w1 = Pµν : {z ∈ C : |z − 1| < 2} → C
Pµν (z) = −sin(πν)π
(z + 1z − 1
)µ2∞∑n=0
Γ(−ν+n)Γ(ν+1+n)n!Γ(1−µ+n)
(1− z
2
)nQµν (z) = − πiπµΓ(1+ν+µ)
2 sin(πµ)Γ(1+ν−µ)P−µν (z) + πeiπµ
2 sin(πµ)Pµν (z)
w2 = Qµν : {z ∈ C : |z| > 1} → C
Qµν (z) = 12eiπµ(z2 − 1)
µ2
∞∑n=0
Γ(ν+µ+22 + n)Γ(ν+µ+1
2 + n)n!Γ(ν + 3
2 + n)z2n
Pµν (z) = e−iπµ sin[π(ν+µ)]π cos(πν) Qµν (z)− e−iπµ sin[π(ν−µ)]
π cos(πν) Qµ−ν−1(z)
would like∂Pµν (z)∂ν
,∂Qµν (z)∂ν
, and∂Pµν (z)∂µ
,∂Qµν (z)∂µ
, for z ∈ C \{−1,1}
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 13 / 31
parameter di�erentiation � associated Legendre functions
domains in C: P µν (z) : |z − 1| < 2 Qµ
ν(z) : |z| > 1
-2 -1 0 1 2 3 4-3
-2
-1
0
1
2
3
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 14 / 31
parameter di�erentiation � associated Legendre functions
general degree derivatives for associated Legendre functions
general degree derivatives for {z ∈ C : |z − 1| < 2}∂∂νP
µν (z) = π cotπνPµν (z)− 1
πAµν (z)
∂∂νQ
µν (z) = π cot(πν)Qµν (z)
− πeiπµΓ(1+ν+µ)2 sin(πµ)Γ(1+ν−µ) [ψ(1 + ν − µ)− ψ(1 + ν − µ)]P−µν (z)
+ eiπµΓ(1+ν+µ)2 sin(πµ)Γ(1+ν−µ)A
−µν (z)− eiπµ
2 sin(πµ)Aµν (z)
∂∂νQ
µν (z) = −π2
2 Pµν (z) + π sin(πµ)
sin(πν) sinπ(ν+µ)Qµν (z)− 1
2 cotπ(ν + µ)Aµν (z)
+12 cscπ(ν + µ)Aµ
ν (−z)
Aµν (z) = sin(πν)
(z+1z−1
)µ2∞∑n=0
Γ(−ν+n)Γ(ν+1+n)n!Γ(1−µ+n)
[ψ(ν+n+1)−ψ(−ν+n)](
1−z2
)nPµν (−z) = e∓iπνPµν (z)− 2
πe−iπµ sin[π(ν + µ)]Qµν (z)
Qµν (−z) = −e±iπνQµν (z)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 15 / 31
parameter di�erentiation � associated Legendre functions
general order derivatives for associated Legendre functions
general order derivatives for {z ∈ C : |z − 1| < 2}∂
∂µPµν (z) =
12Pµν (z) log
z + 1z − 1
− 1πBµν (z)
∂
∂µQµν (z) = − πΓ(1+ν+µ)eiπµ
2Γ(1+ν−µ) sin(πµ) [ψ(1 + ν + µ) + ψ(1 + ν − µ)]P−µν (z)
+π[i− cot(πµ)]Qµν (z) + 12 log z+1
z−1Qµν (z)
+ Γ(1+ν+µ)eiπµ
Γ(1+ν−µ) sin(πµ)B−µν (z)− eiπµ
2 sin(πµ)Bµν (z)
Bµν (z) = sin(πν)
(z + 1z − 1
)µ2∞∑n=0
Γ(ν + n+ 1)ψ(n− µ+ 1)n!Γ(ν − n+ 1)Γ(n− µ+ 1)
(1− z
2
)n
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 16 / 31
parameter di�erentiation � associated Legendre functions
general order derivatives for associated Legendre functions
general order derivatives for {z ∈ C : |z| > 1}∂
∂µQµν (z) = [iπ +
12
log(z2 − 1)]Qµν (z) +14Cµν (z)
∂
∂µPµν (z) = [iπ + 1
2 log(z2 − 1)]Pµν (z)
+ e−2iπµ
cos(πν)
[e−iπνQµν (z) + eiπνQµ−ν−1(z)
]+ e−iπν
4π cos(πν)
{sin[π(ν + µ)]Cµ
ν (z)− sin[π(ν − µ)]Cµ−ν−1(z)
}Cµν (z) = eiπµ(z2 − 1)
µ2
×∞∑n=0
Γ( ν+µ+22
+n)Γ( ν+µ+12
+n)n!Γ(ν+ 3
2+n)z2n
[ψ(ν+µ+2
2 + n) + ψ(ν+µ+12 + n)
]
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 17 / 31
parameter di�erentiation � associated Legendre functions
general degree derivatives for associated Legendre functions
general degree derivatives for {z ∈ C : |z| > 1}∂
∂νQµν (z) =
14Cµν (z)− 1
2Dµν (z)
∂
∂νPµν (z) =
e−iπµ cos(πµ)cos2(πν)
[Qµν (z)−Qµ−ν−1(z)
]+ e−iπµ
4π cos(πν)
{sin[π(ν + µ)]Cµ
ν (z)− sin[π(ν − µ)]Cµ−ν−1(z)
}− e−iπµ
2π cos(πν)
{sin[π(ν + µ)]Dµ
ν (z)− sin[π(ν − µ)]Dµ−ν−1(z)
}Dµν (z) = eiπµ(z2 − 1)
µ2
∞∑n=0
Γ( ν+µ+22
+n)Γ( ν+µ+12
+n)n!Γ(ν+ 3
2+n)z2n ψ(ν + 3
2 + n)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 18 / 31
parameter di�erentiation � associated Legendre functions
What are these Aµν(z), B
µν(z), C
µν(z), D
µν(z)?
most likely independent set of 2 functions
derivatives with respect to parameters of 2F1(a, b; c; z)
∞∑n=0
(a)n(b)nzn
n!(c)nψ(a+ n) = ψ(a)2F1(a, b; c; z) +
∂
∂a2F1(a, b; c; z)
∞∑n=0
(a)n(b)nzn
n!(c)nψ(c+ n) = ψ(c)2F1(a, b; c; z)− ∂
∂c2F1(a, b; c; z)
Brychkov & Geddess (2004) �Di�erentiation of generalized
hypergeometric functions with respect to parameters does not, in
general, lead to generalized hypergeometric functions�
Ancarani & Gasaneo (2008,2009,2010) compute derivatives with
respect to parameters of pFq and show relation to Kampé de Fériet
double hypergeometric series
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 19 / 31
parameter di�erentiation � associated Legendre functions
di�erentiation with respect to parameters of Gausshypergeometric functions in terms of Kampé de Férietmultiple hypergeometric series
∂
∂a2F1(a, b; c; z) =
zb
cF 2:2;1
2:1;0
[a+ 1, b+ 1 : 1, a; 1;2, c+ 1 : a+ 1;−;
z, z
]∂
∂c2F1(a, b; c; z) = −zab
c2F 2:2;1
2:1;0
[a+ 1, b+ 1 : 1, c; 1;2, c+ 1 : c+ 1;−;
z, z
]F p:q;kl:m;n
[(ap) : (bq); (ck);(αl) : (βm); (γn);
x, y
]=
∞∑r,s=0
Qpj=1(aj)r+s
Qqj=1(bj)r
Qkj=1(cj)sQl
j=1(αj)r+sQmj=1(βj)r
Qnj=1(γj)s
xr
r!ys
s!∞∑n=0
(a)n(b)nzn
n!(c)nψ(a+n)=ψ(a)2F1(a,b;c;z)+
zb
cF 2:2;1
2:1;0
[a+ 1, b+ 1 : 1, a; 1;2, c+ 1 : a+ 1;−;
z, z
]∞∑n=0
(a)n(b)nzn
n!(c)nψ(c+n)=ψ(c)2F1(a,b;c;z)+
zab
c2F 2:2;1
2:1;0
24a+ 1, b+ 1 : 1, c; 1;2, c+ 1 : c+ 1;−;
z, z
35Howard Cohl (NIST) parameter di�erentiation April 7, 2011 20 / 31
some particular cases for Legendre functions
Szmytkowski's formulas (see Szmytkowski (2009))
[∂
∂νPmν (z)
]ν=p
= Pmp (z) logz + 1
2
+ [2ψ(2p+ 1)− ψ(p+ 1)− ψ(p−m+ 1)]Pmp (z)
+ (−1)p+mp−m−1∑k=0
(−1)k2k + 2m+ 1
(p−m− k)(p+m+ k + 1)
×[1 +
k!(p+m)!(k + 2m)!(p−m)!
]Pmk+m(z)
+ (−1)p(p+m)!(p−m)!
m−1∑k=0
(−1)k2k + 1
(p− k)(p+ k + 1)P−mk (z)
where p,m ∈ {0, 1, 2, . . .} and 0 ≤ m ≤ p
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 21 / 31
some particular cases for Legendre functions
Szmytkowski's formulas
Some special cases include for m = 0[∂
∂νPν(z)
]ν=p
= Pp(z) logz + 1
2+ 2 [ψ(2p+ 1)− ψ(p+ 1)]Pp(z)
+ 2(−1)pp−1∑k=0
(−1)k2k + 1
(p− k)(p+ k + 1)Pk(z)
and for m = p[∂
∂νP pν (z)
]ν=p
= P pp (z) logz + 1
2+ [2ψ(2p+ 1)− ψ(p+ 1) + γ]P pp (z)
+ (−1)p(2p)!p−1∑k=0
(−1)k2k + 1
(p− k)(p+ k + 1)P−pk (z)
where γ ≈ 0.57721566 is Euler's constant
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 22 / 31
some particular cases for Legendre functions
Application of Szmytkowski's formula
Recently I proved:
‖x− x′‖ν =eiπ(ν+d−1)/2Γ
(d−2
2
)√πΓ(−ν
2
) (r2> − r2
<
)(ν+d−1)/2
(rr′)(d−1)/2
×∞∑λ=0
(λ+
d
2− 1)Q
(1−ν−d)/2λ+(d−3)/2
(r2 + r′2
2rr′
)Cd/2−1λ (cos γ)
for x,x′ ∈ Rd with r = ‖x‖, r′ = ‖x′‖, and r≶ = minmax{r, r
′}, then using
limν→0
∂
∂ν‖x− x′‖ν = ‖x− x′‖2p log ‖x− x′‖
and Whipple's transformation of Legendre functions to obtain a
Gegenbauer expansion for singular logarithmic kernels associated with
fundamental solutions of the even-dimensional polyharmonic equation in
Euclidean spaceHoward Cohl (NIST) parameter di�erentiation April 7, 2011 23 / 31
some particular cases for Legendre functions
Cohl's formulas (see Cohl (2010,2011))
In the sequel m,n ∈ {0, 1, 2, . . .}
Γ(ν ∓m+ 12)
Γ(ν −m+ 12)
[∂
∂µPµν−1/2(z)
]µ=±m
= Qmν−1/2(z) + ψ
(ν ∓m+
12
)Pmν−1/2(z)
±m!m−1∑k=0
(−1)k−m(z2 − 1
)(k−m)/2
2k−m+1k!(m− k)P kν+k−m−1/2(z)
Γ(ν ∓m+ 12)
Γ(ν −m+ 12)
[∂
∂µQµν−1/2(z)
]µ=±m
=[iπ + ψ
(ν ∓m+ 1
2
)]Qmν−1/2(z)
±m!m−1∑k=0
(−1)k−m(z2−1)(k−m)/2
k!(m−k)2k−m+1 Qkν+k−m−1/2(z)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 24 / 31
some particular cases for Legendre functions
Cohl's formulas (see Cohl (2010,2011))
In the sequel m,n ∈ {0, 1, 2, . . .}
±[∂
∂νPµν−1/2(z)
]ν=±n
=[ψ
(µ+ n+
12
)− ψ
(µ− n+
12
)]Pµn−1/2(z)
+n−1∑k=0
n! Γ (µ− n+ 1/2)(z2 − 1
)(n−k)/2
Γ(µ+ n− 2k + 1
2
)k!(n− k)2k−n+1
Pµ+n−kk−1/2 (z)
[∂
∂νQµν−1/2(z)
]ν=±n
= −√π
2eiπµ Γ
(µ− n+
12
)(z2 − 1
)−1/4Qnµ−1/2
(z√
z2 − 1
)
±n!n−1∑k=0
(z2 − 1
)(n−k)/2
2k−n+1k!(n− k)Qµ+k−nk−1/2 (z)
Howard Cohl (NIST) parameter di�erentiation April 7, 2011 25 / 31
references
Bessel references
A. Apelblat and N. Kravitsky.
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references
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Derivatives with respect to the degree and order of associated Legendre functions for|z| > 1 using modi�ed Bessel functions.Integral Transforms and Special Functions, 21(8):581�588, 2010.
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Howard Cohl (NIST) parameter di�erentiation April 7, 2011 30 / 31
summary
Future work in this classical area of Legendre functions
organize the known results, which are scattered, for parameter
derivatives in particular cases
�nd connections and pursue properties of Aµν ,B
µν ,C
µν ,D
µν functions
study multiple derivatives with respect to parameters
investigate anti-derivatives with respect to parameters
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