Abstract Generated abstract
The paper introduces incomplete Macdonald functions defined by splitting the standard integral representation at a finite limit, and notes their occurrence in parabolic partial differential equations for turbulent diffusion and in calculations of scintillation detector response to gamma radiation. It establishes a symmetry relation between the two incomplete functions and derives recurrence formulas that reduce to the classical recurrence relation for Macdonald functions in limiting cases. The authors further express these functions through integro-exponential and incomplete gamma functions, including formulas for arbitrary and positive integer orders, and state convergence of the resulting series for all relevant values of the variables.
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MATHEMATICAL PHYSICS
O. S. BERLYAND, L. V. KIRICHENKO, R. M. KOGAN
ON THE THEORY OF INCOMPLETE MACDONALD FUNCTIONS
(Presented by Academician E. K. Fedorov on 6 VII 1964)
The functions
\[ B_m(x,z)=\frac{1}{2}\int_x^\infty \xi^{m-1} e^{-\frac{1}{2}z\left(\xi+\frac{1}{\xi}\right)}\,d\xi, \tag{1a} \]
\[ K_m(x,z)=\frac{1}{2}\int_0^x \xi^{m-1} e^{-\frac{1}{2}z\left(\xi+\frac{1}{\xi}\right)}\,d\xi \tag{1б} \]
we call incomplete Macdonald functions.
The functions \(B_m(x,z)\) and \(K_m(x,z)\) occur in the solution of partial differential equations of parabolic type, for example, the stationary equation of turbulent diffusion of a radioactive impurity in the atmosphere with allowance for wind transport \((u_z=\mathrm{const})\) from local sources. Then, when the coefficient of turbulence varies with height \((k_z=k_0 z^n;\ 0 \geq n \geq 1)\), the concentration \(q(x,z)=A(z)K_m(x,z)\).
The function \(K_m(x,z)\) also occurs in calculations of the interaction effect of gamma quanta with the scintillator material. In the latter case the recorded signal \(g(E,E_0)\), where \(E\) is the signal amplitude, has the form of a Gaussian curve with variance proportional to the energy of the incident quanta \(E_0\). If the spectrum \(\omega(E_0)\) of the incident radiation is continuous, then the resulting effect is
\[ g(E,E_1)=2\sum_{m=1}^{\infty} a_{m-1}E^{-m}e^{\beta E}K_m(E_1,\beta E). \]
Here \(a_{m-1}\) are the coefficients of the Taylor-series expansion of the product \(\omega(E_0)\rho(E_0)\), where \(\rho(E_0)\) denotes the detector efficiency; \(\beta\) is a constant coefficient depending on the resolving power of the spectrometer.
One could also indicate a number of other applications of incomplete Macdonald functions.
It is easy to show that
\[ K_m(x,z)=B_{-m}\left(\frac{1}{x},z\right). \tag{2} \]
Let us integrate (1) by parts; then we obtain the following recurrence formulas for the functions \(B_m(x,z)\) and \(K_m(x,z)\):
\[ B_{m+1}(x,z)=\frac{2m}{z}B_m(x,z)+B_{m-1}(x,z)+\frac{x^m}{z}e^{-\frac{1}{2}z\left(x+\frac{1}{x}\right)}, \tag{3a} \]
\[ K_{m+1}(x,z)=\frac{2m}{z}K_m(x,z)+K_{m-1}(x,z)-\frac{x^m}{z}e^{-\frac{1}{2}z\left(x+\frac{1}{x}\right)}. \tag{3б} \]
Formulas (3) are valid for an arbitrary number \(m\). In the particular case, from formula (3a) as \(x\to 0\) and from formula (3б) as \(x\to\infty\), the known recurrence formula for Macdonald functions is obtained. The same recurrence formula is obtained from the sum of formulas (3a) and (3б).
1. Consider the function \(B_m(x,z)\) of arbitrary order. We introduce the concept of an integro-exponential function with negative index
\[ E_{-m}(x)=\int_0^\infty u^m e^{-xu}\,du. \tag{4} \]
It is easy to show that
\[ E_{-m}(x)=\frac{1}{x^{m+1}}\Gamma(m+1,x), \]
where
\[ \Gamma(m,x)=\Gamma(m)-\int_0^x u^{m-1}e^u\,du=\Gamma(m)-\gamma(m,x). \]
Here \(\Gamma(m)\) is the gamma function; \(\gamma(m,x)\) is the tabulated incomplete gamma function \({}^{1}\). If in (1) we put \(\xi=xt\), then
\[ B_{m+1}(x,z)=\frac{x^{m+1}}{2}\int_0^\infty t^m e^{-at}\sum_{p=1}^{\infty}\frac{1}{p!}\left(\frac{b}{t}\right)^p\,dt \tag{5} \]
\[ \left(a=-\frac{1}{2}xz;\quad b=-\frac{1}{2}\cdot\frac{z}{x}\right). \]
Integration of (5) for arbitrary \(m\) gives
\[ B_{m+1}(x,z)=\frac{x^{m+1}}{2}\sum_{k=0}^{\infty}\frac{1}{k!}b^k E_{k-m}(a), \tag{6} \]
where
\[ E_m(x)=\int_1^\infty u^{-m}e^{xu}\,du \]
is the tabulated integro-exponential function \({}^{2}\). For \(m<0\)
\[ B_{-m}(x,z)=\frac{x^m}{2}\sum_{k=0}^{\infty}\frac{1}{k!}b^k E_{k+m+1}(a). \tag{7} \]
From formulas (2) and (7) one easily obtains the following expression for the function \(K_m(x,z)\) for \(m>0\):
\[ K_m(x,z)=\frac{x^m}{2}\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}a^k E_{k+m+1}(-b). \tag{8} \]
- For positive integer \(m\), using expression (5), one can obtain the following formula for computing the function \(B_m(x,z)\):
\[ \begin{aligned} B_{m+1}(x,z)=\frac{x^{m+1}}{2}\Biggl\{& m!\,\frac{e^{-a}}{a}\left[ \frac{1}{m!}+\frac{1}{(m-1)!}\left(\frac{1}{a}+\frac{b}{m}\right)+\right.\\ &+\frac{1}{(m-2)!}\left(\frac{1}{a^2}+\frac{b}{am}+\frac{b^2}{2!\,m(m-1)}\right)+\cdots\\ &\left.\cdots+\frac{1}{2!}\left(\frac{1}{a^{m-2}}+\frac{b}{ma^{m-3}}+\cdots+\frac{b^{m-2}}{(m-2)!\,m!/2}\right)+\right.\\ &\left.+\frac{1}{1!}\left(\frac{1}{a^{m-1}}+\frac{b}{ma^{m-2}}+\cdots+\frac{b^{m-1}}{(m-1)!\,m!}\right)+\right.\\ &\left.+\left(\frac{1}{a^m}+\frac{b}{ma^{m-1}}+\cdots+\frac{b^m}{(m!)^2}\right)\right] +\sum_{m=1}^{\infty}\frac{t^k}{k!}E_{k-m}(a) \Biggr\}. \end{aligned} \tag{9} \]
The series (6), (7), (8), and (9) converge for any values of \(x\) and \(z\), owing to the fact that
\[ E_n(y)>E_{n+1}(y). \]
Institute of Applied
Geophysics
Received
17 VI 1964
References
\({}^{1}\) V. I. Pagurova, Tables of the Incomplete Gamma Function, Moscow, 1963.
\({}^{2}\) V. I. Pagurova, Tables of the Integro-Exponential Function, Moscow, 1959.