Generalized derivatives of spherical associated homogeneous distributions on R n

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Generalized derivatives of spherical associated homogeneous distributions on R ⁿ

Ghislain R. Franssens

Belgian Institute for Space Aeronomy, Ringlaan 3, B-1180 Brussels, Belgium

a r t i c l e i n f o a b s t r a c t

Article history:

Received 20 October 2008 Available online 21 February 2009 Submitted by M. Milman Keywords:

Generalized partial derivative Associated homogeneous distribution Potential problem

Full proofs are given of general expressions for the generalized partial derivatives of spherically symmetric associated homogeneous distributions (SAHDs) based on Rⁿ. This work complements earlier work began by Estrada and Kanwal. Special attention is given to the cases when the derivative of the distribution is a singular distribution, being either an analytic continuation or a distributional extension (or regularization). The presented results are useful for the distributional treatment of potential (i.e., time-invariant) problems inn dimensions.

©2009 Elsevier Inc. All rights reserved.

1. Introduction

Homogeneous distributions (HDs) generalize the concept of homogeneous functions, such as

|

^x

|

^z

:

^Rn

→

C, which is a homogeneous function of complex degree z. Associated to homogeneous functions are power-log functions, which arise when taking the derivative with respect to the degree of homogeneity z. The set of associated homogeneous distributions (AHDs) with support in (or based on)Rⁿ, denoted byH

(

Rⁿ

)

, generalizes these power-log functions [5] (or [6]), [11,13,14].

Distributions in H

(

Rⁿ

)

are important for solving distributional potential problems in ndimensions, arising as static (i.e., time-invariant) problems in various branches of physics such as electrostatics, magnetostatics, stationary gravity, molecular theory, etc.

An important subset ofH

(

Rⁿ

)

are the O

(

n

)

-invariant AHDs onRⁿ, called spherical associated homogeneous distributions (SAHDs), and which we denote by SH

(

Rⁿ

)

. A prominent example isr^z, having degree of homogeneity z

∈

^C and order of associationm

=

0, see e.g., [11, pp. 71, 98 and 192]. For a more detailed study of SAHDs, including the singular distributions

((

D^m_z

|

^x

|

^z

)

e

)

z=−ⁿ−^2p,

∀

^m

,

p

∈

N, and how they can be generated as pullbacks of AHDs onR, see [9], [10, Chapter 7]. For an introduction to the more modern concept of extension of a partial distribution, which generalizes the classical concept of regularization of a singular distribution, see [5,6,9]. SAHDs on Rⁿ arise in spherically symmetric potential problems, such as the construction of a fundamental solutiong(i.e., Green’s distribution) for Poisson’s equation

g

= δ

and its complex degree generalizations (i.e., involving “complex powers of the Laplacian

”). Also, proper associated distributions (having order of associationm

>

0) do arise in this context, e.g.,gin

^m+pg

= δ

, in casen

=

^2m,

∀

^m

∈

^Z+and

∀

^p

∈

N, is proportional to the distributionr^2plnr.

A distributional treatment of potential problems imposes itself when the problem involves point sources and one needs to calculate derivatives of potentials at the source point. A function description fails in this case since a potential pro- portional to r^−p for p

>

0, is not differentiable at the origin. To give a proper distributional formulation of these type of problems, expressions for generalized partial derivatives of distributions inSH

(

Rⁿ

)

are required. In this paper, we give gen-

E-mail address:ghislain.franssens@aeronomy.be.

URL:http://www.aeronomy.be.

0022-247X/$ – see front matter ©2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2009.02.031

eral expressions for generalized partial derivatives of SAHDs on Rⁿ. Notice that any monomial combination of generalized partial derivatives of any f

∈

SH

(

Rⁿ

)

is inH

(

Rⁿ

)\SH

(

Rⁿ

)

.

Generalized partial derivatives of the distribution r^−p,

∀

^p

∈

^Z+, were derived in [3] and in [4] (see also [12]). More general expressions, valid for all complex degrees of homogeneity and all orders of association appear not to be available yet. The here presented study fills this gap and so completes (and corrects minor errors in) the work began in [3,4,12]. Our main results are (i) expressions for the derivatives of the singular distributions

((

D^m_z

|

^x

|

^z

)

e

)

z=−ⁿ−^2p,

∀

^m

,

p

∈

N, of which only the casem

=

0 was considered in [3,4], (ii) a careful investigation of the special cases when the distribution itself or its generalized partial derivative is either a regular distribution or a singular distribution that is either an analytic continuation or a distributional extension (in classical terms: a regularization) (cf. cases (a), (b) and (c) of Section 4.2), (iii) the (more demanding) proofs of all formulas and the identities (distributional and others) required for the main proofs. Although it is evident that all main proofs can be given by induction, the non-triviality of this procedure for the singular distributions was already noticed in [3, Section 1], and no proofs were given in [3,4,12].

In [9] it is shown that all SAHDs on Rⁿ are pullbacks T^∗, along the function T

:

^X

=

^Rn

\{

⁰

} →

^Y

=

^{R such that} x

→

^y

= |

^x

|

, of AHDs on R [7,8]. A possible route to calculate generalized partial derivatives for SAHDs onRⁿ that then presents itself is to use the generalized chain rule, involving the ordinary derivatives

∂

T

/∂

xⁱ of the function T and the generalized derivatives of AHDs onR. However, this requires that

∂

T

/∂

xⁱ are smooth functions in the whole ofRⁿ. Unfor- tunately, the functions

∂

T

/∂

xⁱ

=

^xi

/|

^x

|

are not smooth at the origin. For this reason we must resort to a direct calculation, based on the definition of the generalized partial derivation operatorD_i.

In Section 3, we start with distributions for which the generalized partial derivative is a regular distribution. In Section 4, we then consider distributions for which the generalized partial derivative is a singular distribution. In its first subsection, expressions for first degree derivatives are derived and in the second subsection, expressions for derivatives of arbitrary degree are proved.

2. Notation

We use the notation introduced in [5,6]. For convenience, some of these are repeated here.

1. Define 1p^{1 if p} is true, else 1p0. Further, em¹m∈^Ze, hence em

=

^{1 if m} is even, else em

=

0 and similarly, om¹m∈^Zo, henceom

=

^{1 ifm}is odd, elseom

=

^0.

2. The falling factorial polynomial and the rising factorial polynomial are,

∀

^m

∈

N, respectively u_(m)¹m=0

+

¹0<mu

(

u

−

¹

)(

u

−

²

) . . .

u

− (

m

−

¹

)

= Γ (

u

+

¹

)

Γ (

u

+

¹

−

^m

) ,

(1)

u^(m)1m=⁰

+

¹0<mu

(

u

+

¹

)(

u

+

²

) . . .

u

+ (

m

−

¹

)

= Γ (

u

+

^m

)

Γ (

z

) .

(2)

3. Let D

(

Di

∈

^N

, ∀

ⁱ

∈

^Z[¹,n]

)

denote the generalized partial derivation operators with respect to the coordinates x

(

xⁱ

, ∀

ⁱ

∈

^Z[¹,n]

)

. Let k

(

ki

∈

^N

, ∀

ⁱ

∈

^Z[¹,n]

)

be a multi-index and K n

i=¹ki. We will use the following implicit multiplication notation

x^k n i=¹

xⁱki

,

(3)

D^k k

!

n i=¹

D^k_iⁱ

k_i

! ,

(4)

(

2x

)

^l l

!

n i=¹

(

2xⁱ

)

^li

l_i

! ,

(5)

ek−^l

((

k

−

^l

)/

2

)!

n i=1

e_ki−li

((

k_i

−

^li

)/

2

)! .

(6)

Further define k l=⁰

k1

l1=⁰

. . .

kn

ln=⁰

.

(7)

In any dummy multi-indexl

(

li

, ∀

ⁱ

∈

^Z[¹,n]

)

,Lwill be a shorthand forn i=¹li.

4. The surface areaAn−¹of the

(

n

−

¹

)

-dimensional unit sphereSⁿ⁻¹and the volumeVn of then-dimensional unit ballBⁿ it bounds, are given by

An−¹

=

²

π

^n/2

Γ (

n

/

2

) ,

(8)

Vn

=

^An−1

n

= π

^n/2

Γ (

n

/

2

+

¹

) .

(9)

Some identities that are used throughout the paper are,

∀

ⁿ

,

p

,

m

∈

^N, A_2p+2

= (

4

π )

^p+1p

!

(

2p

+

¹

) ! ,

(10)

4^m

( −

ⁿ

/

2

−

^p

)

(m)

A_n+2(p+m)−1

(

4

π )

^p+m

= ( −

¹

)

^mAn+2p−1

(

4

π )

^p

,

(11)

2

(

4

π )

^p−q

A_n+2p+1

A2q+²

=

^Vn+2p

(

4

π )

^p

(

2q

+

¹

)!

q

! .

(12)

3. Partial derivatives being regular distributions

Proposition 1.There holds,

∀

ⁱ

∈

^Z[¹,n],

∀

^k

∈

^Z+and

− (

n

−

^2k

) <

Re

(

z

)

, D^k_i

k

! |

^x

|

^z

=

k l=⁰

(

z

/

2

)

_((k+l)/2) ^ek−l

((

k

−

^l

)/

2

)!

(

2xⁱ

)

^l

l

! |

^x

|

^z−(k+l)

.

(13)

Proof. For eachk

∈

^Z+ and

−(

ⁿ

−

^2k

) <

Re

(

z

)

, all distributions in (13) are regular. From the definition of D_i and by using partial integration it follows that the generalized partial derivative in this case is given by the same rule as the ordinary partial derivative. The proof then immediately follows by induction overk. 2

Corollary 2.Eq.(13)generalizes to, for

−(

ⁿ

−

^2K

) <

Re

(

z

)

, D^k

k

! |

^x

|

^z

=

k l=⁰

(

z

/

2

)

((K+^L)/2) ek−^l

((

k

−

^l

)/

2

)!

(

2x

)

^l

l

! |

^x

|

^z−(K+L)

.

(14)

Proof. Follows trivially from (13). 2

Proposition 3.There holds,

∀

ⁱ

∈

^Z[¹,n],

∀

^k

,

m

∈

^Z+and

−(

ⁿ

−

^2k

) <

Re

(

z

)

, D^k_i

k

!

|

^x

|

^zlnm

|

^x

|

=

m

j=⁰

_m

j

2^m−j

k l=0

P^m_(k⁻_+l^j_{)/2−(m−j)}

(

z

/

2

)

^ek−l

((

k

−

^l

)/

2

) !

(

2xⁱ

)

^l

l

! |

^x

|

^z−(k+l)lnj

|

^x

|

,

(15)

with P^m_q

(

u

)

,

∀

^m

∈

N, the following polynomials in u of degree q, P^m_q

(

u

)

^dm

du^mu₍m+^q)

=

¹0^q

q r=0

(

m

+

^r

) !

r

!

^s

(

m

+

^q

,

m

+

^r

)

u^r

,

(16)

with s

(

k

,

r

)

Stirling numbers of the first kind[1, p. 824].

Proof. For

−

ⁿ

<

Re

(

z

)

,D^m_z

|

^x

|

^zis a regular distribution, soD^m_z

|

^x

|

^z

= |

^x

|

^zlnm

|

^x

|

. By combining the definition of the derivative of a distribution with respect to a parameter, Dz[11, pp. 147–151], with the definition of Di, it easily follows that D^k_iD^m_z

=

D^m_zD^k_i,

∀

^k

,

m

∈

^Z+. Then, from Eq. (13), the formula for themth derivative of a product and the identityu_(p)

=

p

r=⁰s

(

p

,

r

)

u^r for Stirling numbers of the first kinds

(

p

,

r

)

[1, 24.1.3, p. 824], the proposition follows. 2

Corollary 4.Eq.(15)generalizes to,

∀

^m

∈

^Z+and

− (

n

−

^2K

) <

Re

(

z

)

, D^k

k

!

|

^x

|

^zlnm

|

^x

|

=

m

j=0

_m

j

2^m−j

k l=0

P^m_(K⁻₊^j_{L)/2−(m−j)}

(

z

/

2

)

^ek−l

((

k

−

^l

)/

2

) !

(

2x

)

^l

l

! |

^x

|

^z−(K+L)lnj

|

^x

|

.

(17)

Proof. Using D^k_iD^m_z

=

^DmzD^k_i, this immediately follows from (14). 2

By analytic continuation, Eqs. (13), (15) and (14), (17) continue to hold

∀ (

z

+

ⁿ

) ∈

^C

\ (

Z₋

∪

^Z[⁰,2K]

)

.

4. Partial derivatives being singular distributions

The evaluation of distributions that are analytic continuations or extensions require the following projection operator,

∀

n

∈

Z₊and

∀

p

,

q

∈

N,Tⁿ_p,q

:

D(Rⁿ

) →

D(Rⁿ

)

such that

ϕ → ψ =

Tⁿ_p,q

ϕ

, with Tⁿ_p,q

ϕ

(

x

) ϕ (

x

) −

p+q

l=0 (l...l)

l=⁰ 1L=^l

∂

^L

ϕ (∂

x

)

^l

x=⁰ x^l

l

!

1l<p

+

¹p^l1_[+

1

− |

^x

|

²

,

(18)

wherein

(∂

x

)

^l

(∂

x¹

)

^l1

. . . (∂

xⁿ

)

^ln and the step function 1_[+

(

x

) =

1 iffx0. We further defineT_pⁿ_,0,

∀

p

∈

Z₋, as the identity operator onD(^Rn

)

. We will need the following properties of this operator,

∀

ⁱ

∈

^Z[1,n] and

∀ ϕ ∈

D(^Rn

)

,

Tⁿ_p,q

∂ ϕ

∂

xⁱ

= ∂

∂

xⁱ

Tⁿ_p+1,q

ϕ

,

(19)

Tⁿ_p,q

xⁱ

ϕ

=

^xi

Tⁿ_p−1,q

ϕ

.

(20)

Eqs. (19)–(20) are easily verified by direct substitution.

We will need the following distributions,

∀

^p

,

m

∈

^Nand

∀ ϕ _∈

D(Rⁿ

)

. (i) Atz

= −

ⁿ

− (

2p

+

¹

)

,

x^k

|

^x

|

^{−n−(2p+1)}

ln^m

|

^x

| , ϕ

Rⁿ

x^k

|

^x

|

^{−n−(2p+1)lnm}

|

^x

|

T_2pⁿ _+1−K,0

ϕ

ω

Rⁿ

.

(21)

Definition (21) coincides for K

=

0 with the definition for the distributions

|

^x

|

^{−n−(2p+1)lnm}

|

^x

|

given in [9, Eq. (43)]. It is easily verified, by invoking (20), that

x^k

|

^x

|

^{−n−(2p+1)}

ln^m

|

^x

| =

^xk

.

|

^x

|

^{−n−(2p+1)lnm}

|

^x

|

^xk

|

^x

|

^{−n−(2p+1)lnm}

|

^x

|

,

(22)

with the dot in the middle expression denoting multiplication of a smooth function with a distribution. Eq. (21) shows that the distributions

(

x^k

|

^x

|

^{−n−(2p+1)}

)

ln^m

|

^x

|

are analytic continuations for K^2p

+

1 and regular distributions for 2p

+

¹

<

K.

(ii) Atz

= −

ⁿ

−

^2p, D^m_z

x^k

|

^x

|

^z

0

z=−n−2p

, ϕ

Rⁿ

x^k

|

^x

|

^−n−2plnm

|

^x

|

T_2pⁿ _−K,0

ϕ

ω

Rⁿ

.

(23)

Definition (23) coincides for K

=

0 with the definition for the distributions

((

D^m_z

|

^x

|

^z

)

0

)

z=−n−2p given in [9, Eqs. (51) and (18)]. The general associated homogeneous extension is given by

D^m_z

x^k k

! |

^x

|

^z

e

z=−ⁿ−^2p

=

D^m_z x^k

k

! |

^x

|

^z

0

z=−ⁿ−^2p

+

^c

(−

¹

)

^K k l=⁰

1_L2p−K e_k−l

((

k

−

^l

)/

2

) !

(

2D

)

^l l

!

^p−(K+L)/2

(

p

− (

K

+

^L

)/

2

) ! δ,

(24)

whereinc

∈

^Cis arbitrary and D^l

^q

δ, ϕ

=

^qD^l

δ, ϕ

^q

∂

^L

(∂

x

)

^l

ϕ

x=⁰

,

(25)

with

the generalized and ordinary Laplacian onRⁿ, respectively. It is easily verified, by invoking (20) and (78), that also D^m_z

x^k

|

^x

|

^z

e

z=−ⁿ−^2p

=

^xk

.

D^m_z

|

^x

|

^z

e

z=−ⁿ−^2p

^xk D^m_z

|

^x

|

^z

e

z=−ⁿ−^2p

,

(26)

wherein D^m_z

|

^x

|

^z

e

z=−n−2p

=

D^m_z

|

^x

|

^z

0

z=−n−2p

+

^c

^p

p

! δ,

(27)

and c in (27) is the same constant as in (24). Eq. (23) shows that

((

D^m_z

(

x^k

|

^x

|

^z

))

0

)

z=−ⁿ−^2p are particular extensions for K^2p and regular distributions for 2p

<

K. This is also clear from the residue of D^m_z

(

x^k

|

^x

|

^z

)

at z

= −

ⁿ

−

2p, which is easily obtained from the residue of D^m_z

|

^x

|

^z [9, Eq. (48)] and by applying (78), as

z=−Resⁿ−^2p

D^m_z

x^k k

! |

^x

|

^z

= ( −

¹

)

^mAn+2p−1

(

4

π )

^p

( −

¹

)

^K

k l=0

1L^2p−^K e_k−l

((

k

−

^l

)/

2

) !

(

2D

)

^l l

!

^p−(K+L)/2

(

p

− (

K

+

^L

)/

2

) ! δ.

(28)

4.1. First degree derivatives

We now consider the following three remaining cases.

Case (a) refers to the regular distributions

|

^x

|

^−n+1lnm

|

^x

|

^,

∀

^m

∈

N. By the definition of a regular distribution and of Di, in terms of the operatorTⁿ_p,qand after singling out the integration overxⁱ, we obtain,

∀ ϕ _∈

D(Rⁿ

)

,

Di

|

^x

|

^−n+1lnm

|

^x

| , ϕ

_{= −}

Rⁿ⁻¹

+∞

−∞

|

^x

|

^−n+1lnm

|

^x

| ∂

∂

xⁱ

T₀ⁿ_,0

ϕ

i

ω

_Rn−1

.

(29)

Case (b) refers to singular distributions of the form (21) with K

=

0, which are analytic continuations. From (21), the definition of D_i, the property (19) and after singling out the integration overxⁱ, we obtain,

∀ ϕ ∈

D(^Rn

)

,

Di

|

^x

|

^{−n−(2p+1)lnm}

|

^x

| , ϕ

_{= −}

Rⁿ⁻¹ +∞

−∞

|

^x

|

^{−n−(2p+1)lnm}

|

^x

| ∂

∂

xⁱ

T₂ⁿ_(p+1),0

ϕ

dxⁱ

ω

_Rn−1

.

(30)

Case (c) refers to singular distributions of the form (24) with K

=

0, which are extensions. From (23), the definition of Di, the property (19) and after singling out the integration overxⁱ, we obtain,

∀ ϕ _∈

D(Rⁿ

)

,

D_i D^m_z

|

^x

|

^z

0

z=−n−2p

, ϕ

= −

Rⁿ⁻¹ +∞

−∞

|

^x

|

^−n−2plnm

|

^x

| ∂

∂

xⁱ

T_2pⁿ_+1,0

ϕ

dxⁱ

ω

_Rⁿ−1

.

(31)

In each of Eqs. (29), (30) and (31), the inner integral is of the form +∞

−∞

xⁱ

, ρ

^{−n−(k−1)}ln^mxⁱ

, ρ

∂

∂

xⁱ

T_kⁿ_,0

ϕ

dxⁱ

,

(32)

withk

∈ {−

¹

} ∪

N. In order to perform the partial integration of the integral (32), we subdivide the integration regionRⁿ⁻¹ in (31) in the closed ball with radius 0

ρ

1 and its exterior 1

< ρ

. Inside the closed ball, the integration interval of the integral over xⁱ is subdivided in the three subintervals

]−∞ , −

1

− ρ

²

_[

,

[−

1

− ρ

²

, +

1

− ρ

²

_]

and

]+

1

− ρ

²

, +∞[

^{, in} each of which T_kⁿ_,0

ϕ

is continuous in xⁱ. The integral (32) is calculated in Appendix A.1 and the result is given by (58). In Appendix A.2 we further apply the integration

−

Rⁿ⁻¹

ω

_Rn−1 to the boundary term B_k,m

( ρ )

occurring in (58). The result of this integration is given by (60).

Substituting expression (58) for the inner integral (32) and using (60) and (20) yields for the right-hand sides of (29), (30) and (31) the common expression

1_1ko_k1_m=0 V_n+k−1

(

4

π )

^(k−1)/2

^(k−1)/2

((

k

−

¹

)/

2

) !

∂

∂

xⁱ

ϕ

x=0

+

Rⁿ

( −

ⁿ

− (

k

−

¹

))

xⁱ

|

^x

|

^{−n−(k+1)lnm}

|

^x

|

+

^mxi

|

^x

|

^{−n−(k+1)lnm−1}

|

^x

|

^Tnk,0

ϕ

ω

Rⁿ

.

(33) Case (a). We get for (29), from (33) withk

=

^0,

D_i

|

^x

|

^−n+1lnm

|

^x

| , ϕ

=

Rⁿ

c

( −

ⁿ

+

¹

)

xⁱ

|

^x

|

^−n−1lnm

|

^x

|

+

^mxi

|

^x

|

^{−n−1lnm−1}

|

^x

|

^Tn0,0

ϕ

ω

Rⁿ

.

By (21), withki

=

^{1, K}

=

^{1 and p}

=

0, this implies that

D_i

|

^x

|

^−n+1lnm

|

^x

|

= (−

ⁿ

+

¹

)

xⁱ

|

^x

|

⁻ⁿ⁻¹

ln^m

|

^x

| +

^m

xⁱ

|

^x

|

⁻ⁿ⁻¹

ln^m−1

|

^x

|.

⁽³⁴⁾

This shows that the generalized partial derivative Di

( |

^x

|

^−n+1lnm

|

^x

| )

is no longer a regular distribution, but an analytic continuation.

Case (b). We get for (30), from (33) withk

=

²

(

p

+

¹

)

, Di

|

^x

|

^{−n−(2p+1)lnm}

|

^x

| , ϕ

₌

Rⁿ

(−

ⁿ

− (

2p

+

¹

))

xⁱ

|

^x

|

^{−n−(2(p+1)+1)lnm}

|

^x

|

+

^mxi

|

^x

|

^{−n−(2(p+1)+1)lnm−1}

|

^x

|

^T2n(p+1),0

ϕ

ω

Rⁿ

.

By (21), withk_i

=

^{1, K}

=

^{1 and p}replaced byp

+

1, this implies that

Di

|

^x

|

^{−n−(2p+1)lnm}

|

^x

|

= −

n

+ (

2p

+

¹

)

xⁱ

|

^x

|

^{−n−(2p+3)}

ln^m

|

^x

| +

^m

xⁱ

|

^x

|

^{−n−(2p+3)}

ln^m−1

|

^x

| .

(35) This shows that the generalized partial derivative Di

( |

^x

|

^{−n−(2p+1)lnm}

|

^x

| )

is also an analytic continuation.

Case (c). We get for (31), from (33) withk

=

^2p

+

^1, Di

D^m_z

|

^x

|

^z

0

z=−ⁿ−^2p

, ϕ

₌

1m=⁰V_n+2p

(

4

π )

^p

^p

p

!

∂

∂

xⁱ

ϕ

x=⁰

+

Rⁿ

(−

ⁿ

−

^2p

)

xⁱ

|

^x

|

^{−n−2(p+1)lnm}

|

^x

|

+

^mxi

|

^x

|

^{−n−2(p+1)lnm−1}

|

^x

|

^{T2pn +1,0}

ϕ

ω

Rⁿ

.

By (23), withki

=

^{1, K}

=

^{1 and p}replaced byp

+

1, this implies that

Di

D^m_z

|

^x

|

^z

0

z=−ⁿ−^2p

=

¹m=⁰V_n+2p

(

4

π )

^p

^p

p

!

^Di

δ − (

n

+

^2p

)

D^m_zxⁱ

|

^x

|

^z

0

z=−ⁿ−²(p+¹)

+

^m

D^m_z⁻¹xⁱ

|

^x

|

^z

0

z=−ⁿ−²(p+¹)

.

(36)

Eq. (36) reveals that the generalized partial derivative of

((

D^m_z

|

^x

|

^z

)

0

)

z=−ⁿ−^2p contains delta terms ifm

=

0, but not if m

>

0.

Example 1.Forn

=

^{1 andxi}

=

^x,Di

=

^D, (34) yields D

ln^m

|

x

|

=

mx⁻¹ln^m−1

|

x

|,

(37)

wherein the analytic continuation x⁻¹ is Cauchy’s principal value (also written as Pf¹_x). Eq. (37) agrees with [5, Eqs. (143) and (171)], [6, Eqs. (142) and (170)].

Example 2.Form

=

0 follows from (36) that D_i

|

^x

|

₀^−n−2p

= −(

ⁿ

+

^2p

)

xⁱ

|

^x

|

₀^{−n−2(p+1)}

−

^Vn+2p

(

4

π )

^p

^p

p

!

^Di

δ,

(38)

and in particular for p

=

^0,

D_i

|

^x

|

⁻₀ⁿ

= −

^nxi

|

^x

|

⁻₀ⁿ⁻²

−

^VnD_i

δ.

(39)

Letn

=

^{3, put}

−

ⁿ

−

^2p

= − (

k

+

²

)

or 2p

=

^k

−

^{1 (withk}

∈

^Zo,+) in (38) and use the following identity (which is a direct consequence of (10) fork

=

^2p

+

^1),

Vk+²

(

4

π )

^(k−1)/2

((

k

−

¹

)/

2

) ! =

⁴

π

k

+

² 1

k

! ,

(40)

to get

Di

|

^x

|

⁻⁽₀ ^k+2)

= − (

k

+

²

)

xⁱ

|

^x

|

⁻⁽₀ ^k+4)

−

^ok 4

π

k

+

² 1

k

!

^(k−1)/2Di

δ.

(41)

This shows that [12, p. 136, Eq. (16)] is wrong. Further, takingk

=

1 in (41) we get D_i

|

^x

|

⁻₀³

= −

^3xi

|

^x

|

⁻₀⁵

−

⁴

π

3 D_i

δ,

(42)

instead of [12, p. 135, Eq. (14)].

4.2. Higher degree derivatives 4.2.1. The cases (a) and (b)

Proposition 5.There holds,

∀

ⁱ

∈

^Z[1,n],

∀

^k

∈

^Z+,

∀

^m

∈

^Nand

∀

^z

∈ {−

¹

} ∪

^Zo,+, D^k_i

k

!

|

^x

|

^−n−zlnm

|

^x

|

=

m

j=0

_m

j

2^m−j

k l=0

P₍^m_k⁻_+l^j_{)/2−(m−j)}

(

z

/

2

)

^ek−l

((

k

−

^l

)/

2

) !

(

2xⁱ

)

^l

l

! |

^x

|

^{−n−z−(k+l)}

ln^j

|

^x

| ,

(43)

wherein the distributions in the right-hand side are analytic continuations.

Proof. It follows from (34) and (35) that the generalized derivative of

|

^x

|

^−n−zlnm

|

^x

|

^,

∀

^m

∈

^Nandz

∈ {−

¹

} ∪

^Zo,+, is given by Leibniz’ rule. Hence the proof of (43) is identical to the one leading to (15). 2