# Basic Properties of Incomplete Macdonald Function with Applications

Authors: Jian-Jun Shu; Kunal Krishnaraj Shastri
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the basic properties of the incomplete Macdonald function, such as recurrence and differential relations and series and asymptotic expansions. This paper also shows that the incomplete Macdonald function, as a simple closed-form expression, is a particular solution to a parabolic partial differential equation, which arises naturally in a wide variety of transient and diffusion-related phenomena.


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## Body

## 1. Introduction

As a particular solution to the modified Bessel equation [1], the Macdonald function [2] (also known as the modified Bessel function of the second kind, the Basset function, or the modified Hankel function) has been employed in wide-ranging fields to provide analytical solutions for many physical phenomena. The Macdonald function can be expressed in the form of integrals. A new class of incomplete special function, called the incomplete Macdonald function, is defined by having the variable endpoint of integration and arises in wide-ranging contexts such as viscous flow [3], heat conduction [4], groundwater hydrology [5], electromagnetism [6], galaxy [7], nuclear reactor [8], and doubly special relativity [9].The several distinct incomplete versions of the Macdonald function are possible since various integral representations of the Macdonald function often seem unrelated under the variable endpoint of integration. In literature, the specific definition chosen among various forms of the incomplete Macdonald function is directed by the particular application. These various forms and interrelationships of the incomplete Macdonald function are summarized in Table1.Table 1
Various distinct incomplete versions of the Macdonald function.
FunctionDefinitionShu function [3]Sνz,t=12z2ν∫0te−τ−z2/4ττν+1dτGeneralized incomplete gamma function [4]Γν,t;z=∫t∞τν−1e−τ−z/τdτ=2zν/2Sν2z,ztLeaky aquifer function [5]Lνz,t=∫1∞e−zτ−t/ττν+1dτ=2ztν/2S−ν2zt,tIncomplete modified Bessel function [6]1/2∫t∞e−zcoshτcoshντdτ=1/2Sν+S−νz,ze−t/2The difference in the names assigned to various forms of the incomplete Macdonald function is worth noting (as listed in Table1). They are the Shu function of viscous flow, generalized incomplete gamma function of heat conduction, leaky aquifer function of groundwater hydrology, and incomplete modified Bessel function of electromagnetism. It is pretty apparent that there is a lack of communication among different research communities. The Shu function was first employed by Shu and Chwang [3] in the expression of the hydrodynamic force acting on a rigid circular cylinder translating in a time-dependent rotating flow field. The motivation of studying the generalized incomplete gamma function [4] was the role it played in the closed-form solution to several problems in heat conduction. Groundwater hydrologists commonly refer to the leaky aquifer integral as the leaky aquifer function [5], which is useful for determining the hydraulic properties of leaky-confined aquifers. The incomplete modified Bessel function [6] was introduced to express the solution of electromagnetic problems in truncated cylindrical structures. It is worth mentioning at this end that these Shu function, generalized incomplete gamma function, leaky aquifer function, and incomplete modified Bessel function have a well-deserved reputation of being a computational challenge due to their integral representations. The increasing number of applications calls for a rigorous analysis of various forms of the incomplete Macdonald function. To avoid redundancy, the special function analyzed in this paper is mainly referred to as the Shu function, which is a representative of a large class of time-dependent problems.In this paper, the key properties of the Shu function, such as recurrence and differential relations and series and asymptotic expansions, are derived. It is also shown that the Shu function is a particular solution to a parabolic partial differential equation (PDE), which occurs in various transient problems, for example, transient flow in porous media [10], electromagnetic waves in a cylindrical waveguide [11, 12], and diffusion-related phenomena [8].


## 2. Definition

The integral form of the Macdonald function is given [13] by(1)Kνz=12z2ν∫0∞e−τ−z2/4ττν+1dτ.As an incomplete version of the Macdonald function, the definition of the Shu function [3] is adopted for investigation in this paper and obtained by restricting the upper endpoint of the integral in (1) to t,(2)Sνz,t=12z2ν∫0te−τ−z2/4ττν+1dτ,Ret>0andRez>0,in which Re denotes the real part of a complex number, the symbols ν, z, and t are, respectively, termed the order, argument, and endpoint of the Shu function and are generally taken to be complex quantities. Hereafter, the symbols n and x>0 are, respectively, annotated to denote integer order ν and real argument z. The behavior of the Shu function Snx,t of integer order n, real argument x>0, and real endpoint t>0 is depicted in Figures 1 and 2. In view of that the Shu function has infinite when Rez=0, we restrict attention to the complex argument Rez>0 and the complex endpoint Ret>0.Figure 1
Snx,t against real endpoint t>0 for various values of integer order n at x=3.Figure 2
Snx,t against real argument x>0 for various values of integer order n at t=3.By using (1), the Shu function can be written as(3)Sνz,t=Kνz−12z2ν∫t∞e−τ−z2/4ττν+1dτ.Substitutingew=z/2τ in (2), we get [14](4)Sνz,t=12∫lnz/2t∞e−zcoshw+νwdw.By using the substitution:y=z2/4τ in (2), we get the alternate form(5)Sνz,t=122zν∫z2/4t∞yν−1e−y−z2/4ydy.This can be expressed in the form of the generalized incomplete gamma function [4] (Table 1) as(6)Sνz,t=122zνΓν,z24t;z24.Furthermore, the Shu function can be related to the leaky aquifer function [5] (Table 1) by(7)Sνz,t=12z2tνL−νz24t,t.Using this relation, the algorithm [15] proposed for the computation of the numerical value of Lνz,t can be used for the computation of Sνz,t.


## 3. Recurrence Relations

Two recurrence relations of the Shu function are derived in this section.


### 3.1. The First Recurrence Relation

Integrating (2) by parts, we obtain(8)Sνz,t=−12z2νe−τ−z2/4τντν0+t+12z2ν∫0te−τ−z2/4τντν−1+z24τ2dτ.Here, note that(9)∂Sν−1∂t=12z2ν−1e−t−z2/4ttν.Hence, we obtain(10)Sν=−z2ν∂Sν−1∂t−z2νSν−1+z2νSν+1.On simplifying this expression, we obtain the first recurrence relation,(11)−2νzSν=∂Sν−1∂t+Sν−1−Sν+1.


### 3.2. The Second Recurrence Relation

We differentiate (2) with respect to z,(12)∂Sν∂z=ν4z2ν−1∫0te−τ−z2/4ττν+1dτ−12z2ν+1∫0te−τ−z2/4ττν+2dτ.Here,(13)∫0te−τ−z2/4ττν+1dτ=−e−t−z2/4tνtν+∫0te−τ−z2/4τντν−1+z24τ2dτ.Hence, we obtain(14)∂Sν∂z=−14z2ν−1e−t−z2/4ttν+14z2ν−1∫0te−τ−z2/4ττν−1+z24τ2dτ−12z2ν+1∫0te−τ−z2/4ττν+2dτ.On simplifying, we obtain the second recurrence relation,(15)−2∂Sν∂z=∂Sν−1∂t+Sν−1+Sν+1.


## 3.1. The First Recurrence Relation

Integrating (2) by parts, we obtain(8)Sνz,t=−12z2νe−τ−z2/4τντν0+t+12z2ν∫0te−τ−z2/4τντν−1+z24τ2dτ.Here, note that(9)∂Sν−1∂t=12z2ν−1e−t−z2/4ttν.Hence, we obtain(10)Sν=−z2ν∂Sν−1∂t−z2νSν−1+z2νSν+1.On simplifying this expression, we obtain the first recurrence relation,(11)−2νzSν=∂Sν−1∂t+Sν−1−Sν+1.


## 3.2. The Second Recurrence Relation

We differentiate (2) with respect to z,(12)∂Sν∂z=ν4z2ν−1∫0te−τ−z2/4ττν+1dτ−12z2ν+1∫0te−τ−z2/4ττν+2dτ.Here,(13)∫0te−τ−z2/4ττν+1dτ=−e−t−z2/4tνtν+∫0te−τ−z2/4τντν−1+z24τ2dτ.Hence, we obtain(14)∂Sν∂z=−14z2ν−1e−t−z2/4ttν+14z2ν−1∫0te−τ−z2/4ττν−1+z24τ2dτ−12z2ν+1∫0te−τ−z2/4ττν+2dτ.On simplifying, we obtain the second recurrence relation,(15)−2∂Sν∂z=∂Sν−1∂t+Sν−1+Sν+1.


## 4. Differential Relations

Two differential relations of the Shu function are derived in this section.


### 4.1. The First Differential Relation

Adding the two recurrence relations (11) and (15), we obtain(16)−∂Sν∂z−νzSν=Sν−1+∂Sν−1∂t.Premultiplying byzν, we get(17)∂∂zzνSν=−1+∂∂tzνSν−1.This differential relation may be extended as(18)1z∂∂zkzνSν=−1k1+∂∂tkzν−kSν−k,k∈0,1,2,⋯.


### 4.2. The Second Differential Relation

We subtract the second recurrence relation (15) from (11),(19)−∂Sν∂z+νzSν=Sν+1.Premultiplying byz−ν,(20)∂∂zSνzν=−Sν+1zν,or(21)1z∂∂zkSνzν=−1kSν+kzν+k,k∈0,1,2,⋯.


## 4.1. The First Differential Relation

Adding the two recurrence relations (11) and (15), we obtain(16)−∂Sν∂z−νzSν=Sν−1+∂Sν−1∂t.Premultiplying byzν, we get(17)∂∂zzνSν=−1+∂∂tzνSν−1.This differential relation may be extended as(18)1z∂∂zkzνSν=−1k1+∂∂tkzν−kSν−k,k∈0,1,2,⋯.


## 4.2. The Second Differential Relation

We subtract the second recurrence relation (15) from (11),(19)−∂Sν∂z+νzSν=Sν+1.Premultiplying byz−ν,(20)∂∂zSνzν=−Sν+1zν,or(21)1z∂∂zkSνzν=−1kSν+kzν+k,k∈0,1,2,⋯.


## 5. Partial Differential Equation

Replacingν by ν+1 in (16), we obtain(22)−zSν=z∂Sν∂t+z∂Sν+1∂z+ν+1Sν+1.On differentiating (19) with respect to z, we get(23)ν∂Sν∂z=Sν+1+z∂Sν+1∂z+∂Sν∂z+z∂2Sν∂z2.From (19), (22), and (23), it is a straightforward exercise to obtain the following PDE:(24)z2∂2Sν∂z2+z∂Sν∂z−z2+ν2Sν−z2∂Sν∂t=0.The PDE can also be written as(25)Ω−∂∂tSν=0,where Ω represents the modified Bessel operator:(26)Ω≡1z∂∂zz∂∂z−1+ν2z2.This elegant parabolic PDE (25) arises naturally in a wide variety of transient and diffusion-related phenomena.


## 6. Series Expansions

Two expansions ast⟶0 and z⟶0, respectively, are derived in this section, in which  denotes the modulus of a complex number.


### 6.1. Series Expansion ast⟶0

We substitute the expansione−z2/4y=∑k=0∞−1k/k!z/22k1/yk as y⟶∞ in (5) to obtain(27)Sνz,t=∑k=0∞−1k2k!2zν−2k∫z2/4t∞yν−k−1e−ydy,or(28)Sνz,t=∑k=0∞−1k2k!2zν−2kΓν−k,z24t.The asymptotic expansion of the incomplete gamma functionΓν,t=∫t∞τν−1e−τdτ for large t is given [4] by(29)Γν,t=tν−1e−t∑m=0∞−1m1−νm1tm,where νm≡Γm+ν/Γν is the Pochhammer polynomial and Γν=Γν,0=∫0∞τν−1e−τdτ is the gamma function. We substitute the expansion (28) to get(30)Sνz,t=∑m=0∞∑k=0∞−1m+k2k!1−ν+kmz2ν−2m−2e−z2/4ttν−m−k−1.The leading term approximation is given by(31)Sνz,t~12z2ν−2e−z2/4ttν−11+Otast⟶0.The leading term approximation to the actual value of the Shu functionSnx,t of integer order n and real argument x>0 for small endpoint t is shown in Figure 3. As can be observed, the agreement is better for higher integer order n.Figure 3
Leading term approximation fort⟶0+.


### 6.2. Series Expansion asz⟶0

We substitute the expansione−z2/4τ=∑k=0∞−1k/k!1/τkz/22k as z⟶0 in (3) to get(32)Sνz,t=Kνz−∑k=0∞−1k2k!z2ν+2k∫t∞e−ττν+k+1dτ.This expansion can be expressed in a more compact form by using the incomplete gamma functionΓν,t:(33)Sνz,t=Kνz−∑k=0∞−1kΓ−ν−k,t2k!z2ν+2k.By using the approximation [4] of Kνz for small z,(34)Kνz~−lnz1+O1lnz,ν=0,2νsgnReν−1ΓνsgnReνzνsgnReν1+Oz2lnz+Oz2Reν,ν≠0,asz⟶0,in which the signum function(35)sgny=1y>00y=0−1y<0=2π∫0∞sinyττdτ.The leading term approximation is given by(36)Sνz,t~−lnz1+O1lnz,ν=0,2νsgnReν−1ΓνsgnReνzνsgnReν1+Oz2lnz+Oz2Reν,ν≠0,asz⟶0,which agrees with the known result by Shu and Chwang [3] for the case of ν=0. The leading term approximation to the actual value of the Shu function Snx,t of integer order n and real endpoint t>0 for small argument x is shown in Figure 4. As can be observed, the agreement is better for lower integer order n.Figure 4
Leading term approximation forx⟶0+.


## 6.1. Series Expansion ast⟶0

We substitute the expansione−z2/4y=∑k=0∞−1k/k!z/22k1/yk as y⟶∞ in (5) to obtain(27)Sνz,t=∑k=0∞−1k2k!2zν−2k∫z2/4t∞yν−k−1e−ydy,or(28)Sνz,t=∑k=0∞−1k2k!2zν−2kΓν−k,z24t.The asymptotic expansion of the incomplete gamma functionΓν,t=∫t∞τν−1e−τdτ for large t is given [4] by(29)Γν,t=tν−1e−t∑m=0∞−1m1−νm1tm,where νm≡Γm+ν/Γν is the Pochhammer polynomial and Γν=Γν,0=∫0∞τν−1e−τdτ is the gamma function. We substitute the expansion (28) to get(30)Sνz,t=∑m=0∞∑k=0∞−1m+k2k!1−ν+kmz2ν−2m−2e−z2/4ttν−m−k−1.The leading term approximation is given by(31)Sνz,t~12z2ν−2e−z2/4ttν−11+Otast⟶0.The leading term approximation to the actual value of the Shu functionSnx,t of integer order n and real argument x>0 for small endpoint t is shown in Figure 3. As can be observed, the agreement is better for higher integer order n.Figure 3
Leading term approximation fort⟶0+.


## 6.2. Series Expansion asz⟶0

We substitute the expansione−z2/4τ=∑k=0∞−1k/k!1/τkz/22k as z⟶0 in (3) to get(32)Sνz,t=Kνz−∑k=0∞−1k2k!z2ν+2k∫t∞e−ττν+k+1dτ.This expansion can be expressed in a more compact form by using the incomplete gamma functionΓν,t:(33)Sνz,t=Kνz−∑k=0∞−1kΓ−ν−k,t2k!z2ν+2k.By using the approximation [4] of Kνz for small z,(34)Kνz~−lnz1+O1lnz,ν=0,2νsgnReν−1ΓνsgnReνzνsgnReν1+Oz2lnz+Oz2Reν,ν≠0,asz⟶0,in which the signum function(35)sgny=1y>00y=0−1y<0=2π∫0∞sinyττdτ.The leading term approximation is given by(36)Sνz,t~−lnz1+O1lnz,ν=0,2νsgnReν−1ΓνsgnReνzνsgnReν1+Oz2lnz+Oz2Reν,ν≠0,asz⟶0,which agrees with the known result by Shu and Chwang [3] for the case of ν=0. The leading term approximation to the actual value of the Shu function Snx,t of integer order n and real endpoint t>0 for small argument x is shown in Figure 4. As can be observed, the agreement is better for lower integer order n.Figure 4
Leading term approximation forx⟶0+.


## 7. Asymptotic Expansions

Two asymptotic expansions for both larget with fixed z and large z with fixed t, respectively, are derived in this section.


### 7.1. Asymptotic Expansion ast⟶∞

We use the expansions (29) and (33) to get(37)Sνz,t=Kνz−∑m=0∞∑k=0∞−1m+kν+k+1m2k!z2ν+2ke−ttν+m+k+1.The leading term approximation is given by(38)Sνz,t~Kνz1+Oe−Rett1+Reνast⟶∞.The leading term approximation to the actual value of the Shu functionSnx,t of integer order n and real argument x>0 for large endpoint t is shown in Figure 5. As can be observed, the agreement is excellent for all integer order n.Figure 5
Leading term approximation fort⟶+∞.


### 7.2. Asymptotic Expansion asz⟶∞

We consider an integral of the formfz=∫abe−zpwqwdw. If Repc>Repa, ∀c∈a,b, and qw=∑k=0∞qkw−ak, dpw/dw=∑k=0∞k+1pkw−ak, then by the generalization of Laplace’s method [16] for complex integration,(39)fz=e−zpa∑k=0∞Γk+1ckzk+1,where the first two coefficients c0 and c1 are given by(40)c0=q0p0,c1=p0q1−2p1q0p03.We use (4) to derive the asymptotic expansion for large z and note that qw=eνw and pw=coshw. The function Repw=Recoshw attains minima at ζ=a=lnz/2t. Hence, qw=eνw~eνζ+νeνζw−ζ+Ow−ζ2 and dpw/dw=sinhw~sinhζ+coshζw−ζ+Ow−ζ2. Following (39), the asymptotic expansion as z⟶+∞ with fixed t can be easily derived:(41)Sνz,t~eνζ−zcoshζ2zsinhζ1+O1z.The leading term approximation is given by(42)Sνz,t~zνe−z2/4t−t2tν−1z2−4t21+O1zasz⟶∞.It is worth mentioning at this end that the incomplete modified Bessel function,(43)12∫t∞e−zcoshτcoshντdτ=12Sν+S−νz,ze−t2~coshνte−zcosht2zsinht1+O1zasz⟶∞,which agrees with the result of Cicchetti and Faraone [6], but their expression is extremely complicated. The leading term approximation to the actual value of the Shu function Snx,t of integer order n and real endpoint t>0 for large argument x is shown in Figure 6. As can be observed, the agreement is better for lower integer order n.Figure 6
Leading term approximation forx⟶+∞.


## 7.1. Asymptotic Expansion ast⟶∞

We use the expansions (29) and (33) to get(37)Sνz,t=Kνz−∑m=0∞∑k=0∞−1m+kν+k+1m2k!z2ν+2ke−ttν+m+k+1.The leading term approximation is given by(38)Sνz,t~Kνz1+Oe−Rett1+Reνast⟶∞.The leading term approximation to the actual value of the Shu functionSnx,t of integer order n and real argument x>0 for large endpoint t is shown in Figure 5. As can be observed, the agreement is excellent for all integer order n.Figure 5
Leading term approximation fort⟶+∞.


## 7.2. Asymptotic Expansion asz⟶∞

We consider an integral of the formfz=∫abe−zpwqwdw. If Repc>Repa, ∀c∈a,b, and qw=∑k=0∞qkw−ak, dpw/dw=∑k=0∞k+1pkw−ak, then by the generalization of Laplace’s method [16] for complex integration,(39)fz=e−zpa∑k=0∞Γk+1ckzk+1,where the first two coefficients c0 and c1 are given by(40)c0=q0p0,c1=p0q1−2p1q0p03.We use (4) to derive the asymptotic expansion for large z and note that qw=eνw and pw=coshw. The function Repw=Recoshw attains minima at ζ=a=lnz/2t. Hence, qw=eνw~eνζ+νeνζw−ζ+Ow−ζ2 and dpw/dw=sinhw~sinhζ+coshζw−ζ+Ow−ζ2. Following (39), the asymptotic expansion as z⟶+∞ with fixed t can be easily derived:(41)Sνz,t~eνζ−zcoshζ2zsinhζ1+O1z.The leading term approximation is given by(42)Sνz,t~zνe−z2/4t−t2tν−1z2−4t21+O1zasz⟶∞.It is worth mentioning at this end that the incomplete modified Bessel function,(43)12∫t∞e−zcoshτcoshντdτ=12Sν+S−νz,ze−t2~coshνte−zcosht2zsinht1+O1zasz⟶∞,which agrees with the result of Cicchetti and Faraone [6], but their expression is extremely complicated. The leading term approximation to the actual value of the Shu function Snx,t of integer order n and real endpoint t>0 for large argument x is shown in Figure 6. As can be observed, the agreement is better for lower integer order n.Figure 6
Leading term approximation forx⟶+∞.


## 8. Conclusions

A number of key properties of the incomplete Macdonald function have been derived, comprising recurrence and differential relations and series and asymptotic expansions. As can be observed graphically, the agreement is found between the actual value and the corresponding leading term approximation for all limiting cases. It is also shown that the incomplete Macdonald function is a particular solution to the parabolic PDE associated with a wide variety of transient natural phenomena. The Shu function and other distinct incomplete versions of the Macdonald function can be used to find simple closed-form expressions for various natural phenomena.



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