Application of Continued Fractions to the Evaluation of Certain Types of Integrals
Abstract Generated abstract
The paper develops continued fraction methods for approximating integrals of the form \(\int_x^\infty t^{\gamma-1}e^{-ct}\,dt\), where \(c=a+bi\), \(a>0\), \(b>0\), and \(0\leq\gamma<1\). Starting from repeated integration by parts, it converts the resulting asymptotic series into a continued fraction, derives explicit formulas for the even convergents and their numerators, and obtains a computable rational approximation with a bound for the remainder. By separating real and imaginary parts, the result yields approximation formulas and error estimates for related sine and cosine integrals, extending a known real-variable estimate to the complex parameter case.
Full Text
APPLICATION OF CONTINUED FRACTIONS TO THE CALCULATION OF CERTAIN TYPES OF INTEGRALS
V. E. KORNILOV
(Presented by the scientific seminar of the Departments of Higher Mathematics and Engineering-Computational Mathematics of TPI)
In the article the parameters \(a>0,\ b>0,\ 0\leq \gamma<1;\ x>0\) are used,
\[ c=a+bi,\qquad \varphi=\operatorname{arctg}\frac{b}{a}, \]
where
\[ a=|c|\cos\varphi,\qquad b=|c|\sin\varphi,\qquad 0\leq \varphi<\frac{\pi}{2}. \tag{1} \]
The notation is introduced
\[ (\lambda)_k=\lambda(\lambda+1)\ldots(\lambda+k-1)=\frac{\Gamma(\lambda+k)}{\Gamma(\lambda)},\qquad (\lambda)_0=1. \]
Consider the application of continued fractions to the approximate calculation of integrals of the form
\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt. \]
After \(n\)-fold integration by parts we obtain
\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt =x^\gamma e^{-cx}\sum_{k=0}^{n-1}(-1)^k \frac{(1-\gamma)_k}{(cx)^{k+1}} + \]
\[ +(-1)^{n-1}\frac{(1-\gamma)_n}{c^n} \int_x^\infty t^{\gamma-n-1}e^{-ct}\,dt, \]
whence, as \(n\to\infty\), a series for \(I\) is obtained:
\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt =x^\gamma e^{-cx}\sum_{k=0}^{\infty}(-1)^k \frac{(1-\gamma)_k}{(cx)^{k+1}} = \]
\[ =\frac{x^\gamma e^{-cx}}{cx}\, F\left(1-\gamma,-\frac{1}{cx}\right), \qquad |cx|\gg 1, \]
moreover, it is not difficult to verify that
\[ F\left(1-\gamma,-\frac{1}{cx}\right) = 1-\frac{1-\gamma}{cx} +\frac{(1-\gamma)_2}{(cx)^2} F\left(3-\gamma,-\frac{1}{cx}\right). \]
For the series
\[ \frac{1}{cx}F\left(3-\gamma,-\frac{1}{cx}\right) \]
the corresponding continued fraction ([1], p. 229) is the following ([1], pp. 301, 269):
\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt =x^\gamma e^{-cx}\frac{1}{cx}\left\{1-\frac{1-\gamma}{cx}+ \right. \]
\[ \left. +\frac{(1-\gamma)_2}{cx}\left[\frac{P_{2k}(cx)}{Q_{2k}(cx)}+R_{2k}(cx)\right]\right\} =\frac{x^\gamma e^{-cx}}{(cx)^2}\cdot \frac{p_{2k}(cx)}{Q_{2k}(cx)}+r_{2k}(cx), \]
where
\[ \frac{P_n(cx)}{Q_n(cx)} =\frac{1}{a_1+\ldots+\frac{1}{a_n}}, \]
\[ R_{2k}(cx)=\sum_{n=k}^{\infty} \frac{a_{2n+2}}{Q_{2n}(cx)Q_{2n+2}(cx)} \quad ([2], p. 34), \]
\[ a_1=cx,\qquad a_{2k}=\frac{(k-1)!}{(3-\gamma)_k},\qquad a_{2k+1}=\frac{(3-\gamma)_k}{k!}\,cx;\quad k=1,2,\ldots \tag{2} \]
For the denominators of the convergents with even indices the recurrence relation is known ([2], p. 14)
\[ Q_{2k+2}(cx)= \left(a_{2k+2}a_{2k+1}+\frac{a_{2k+2}}{a_{2k}}+1\right)Q_{2k}(cx) -\frac{a_{2k+2}}{a_{2k}}Q_{2k-2}(cx). \tag{3} \]
Next it is assumed that
\[ Q_{2k}(cx)=\sum_{n=0}^{k} C_k^n\frac{(cx)^n}{(3-\gamma)_n},\quad k=1,2,\ldots \tag{4} \]
After a direct computation of the continued fraction (2) we verify that equality (4) holds for \(k=1,2\). On the basis of the method of mathematical induction it is not difficult to establish that if the polynomials (4) satisfy relation (3), then they will also be valid for \(k=3,4,\ldots\)
On the basis of (2)—(4) we obtain
\[ Q_{2k+2}(cx)= \left(\frac{cx}{3-\gamma+k}+\frac{k}{3-\gamma+k}+1\right) \sum_{n=0}^{k}C_k^n\frac{(cx)^n}{(3-\gamma)_n} - \]
\[ -\frac{k}{3-\gamma+k} \sum_{n=0}^{k-1}C_{k-1}^n\frac{(cx)^n}{(3-\gamma)_n} = \]
\[ =1+\frac{k}{3-\gamma+k}\cdot \frac{(cx)^k}{(3-\gamma)_{k-1}} + \frac{(cx)^{k+1}}{(3-\gamma)_{k+1}} + \frac{(cx)^k}{(3-\gamma)_k} + \]
\[ +k\frac{(cx)^k}{(3-\gamma)_{k+1}} +\sum_{n=1}^{k}\frac{(cx)^n}{(3-\gamma)_{n-1}} \left[ \frac{1}{3-\gamma+k}C_k^{\,n-1} + \right. \]
\[ \left. +\frac{1}{2-\gamma+n}C_k^n +\frac{k}{(2-\gamma+n)(3-\gamma+k)}C_{k-1}^{\,n-1} \right] = \]
\[ =1+\sum_{n=1}^{k-1}C_{k+1}^n\frac{(cx)^n}{(3-\gamma)_n} +(k+1)\frac{(cx)^k}{(3-\gamma)_k} + \]
\[ +\frac{(cx)^{k+1}}{(3-\gamma)_{k+1}} = \sum_{n=0}^{k+1}C_{k+1}^n\frac{(cx)^n}{(3-\gamma)_n}, \]
Thus equality (4) is proved.
Moreover, in view of (2), the numerator \(P_{2k}(cx)\) will be as follows:
\[ P_{2k}(cx)=\sum_{n=1}^{k} C_k^n \frac{(cx)^n}{(3-\gamma)_n} \sum_{m=0}^{n-1}(-1)^m \frac{(3-\gamma)_m}{(cx)^{m+1}} . \tag{5} \]
Next, after applying equalities (2), (4), (5), the numerator \(p_{2k}(cx)\) is calculated and transformed:
\[ \begin{aligned} p_{2k}(cx) &=(1-\gamma)_2 P_{2k}(cx)+(cx-1+\gamma)Q_{2k}(cx)= \\ &=(1-\gamma)_2\sum_{n=0}^{k} C_k^n \sum_{m=0}^{n+1}\frac{(-1)^m (cx)^{n+1-m}} {(1-\gamma+m)_{n+2-m}}= \\ &=(1-\gamma)_2\Bigg\{C_k^0\left[\frac{cx}{(1-\gamma)_2}-\frac{1}{2-\gamma}\right]+ \\ &\qquad\qquad +C_k^1\left[\frac{(cx)^2}{(1-\gamma)_3} -\frac{cx}{(2-\gamma)_2}+\frac{1}{3-\gamma}\right]+\cdots \\ &\qquad\qquad \cdots+C_k^k\left[\frac{(cx)^{k+1}}{(1-\gamma)_{k+2}} -\cdots+(-1)^{k+1}\frac{1}{2-\gamma+k}\right]\Bigg\}, \end{aligned} \]
\[ p_{2k}(cx)=(1-\gamma)_2\sum_{n=0}^{k+1}(cx)^n \sum_{m=0}^{k+1-n} \frac{C_k^m(-1)^{k+1-n-m}} {(2-\gamma+k-n-m)_{n+1}} . \tag{6} \]
Taking into account equalities (2), (4), (6), we obtain
\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt = \]
\[ = \frac{ \dfrac{(1-\gamma)_2 x^\gamma}{e^{cx}} \sum_{n=1}^{k+1}(cx)^n \sum_{m=0}^{k+1-n} \dfrac{C_k^m(-1)^{k+1-n-m}} {(2-\gamma+k-n-m)_{n+1}} }{ \sum_{n=0}^{k} C_k^n \dfrac{(cx)^{n+2}}{(3-\gamma)_n} } +r_{2k}(cx),\qquad k=1,2,\ldots . \tag{7} \]
Next, according to (2) and (4) and in view of
\(|Q_{2k}(cx)|<|Q_{2k+2}(cx)|\) ([2], p. 14), the remainder term \(R_{2k}(cx)\) is estimated in modulus:
\[ |R_{2k}(cx)|\le \sum_{n=k}^{\infty} \left| \frac{n!}{(3-\gamma)_{n+1}Q_{2n}(cx)Q_{2n+2}(cx)} \right|< \]
\[ < \frac{k!}{(3-\gamma)_{k-1}|Q_{2k}(cx)|^2} \sum_{n=0}^{\infty} \frac{(k+1)_n}{(2-\gamma+k)_{n+2}}< \]
\[ < \frac{k!}{(3-\gamma)_{k-1}|Q_{2k}(cx)|^2} \sum_{n=0}^{\infty} \frac{1}{(2-\gamma+k+n)_2} = \frac{k!}{(3-\gamma)_k|Q_{2k}(cx)|^2}, \]
or, in view of (2),
\[ |r_{2k}(cx)|< \frac{(1-\gamma)_2 k!\,|x^\gamma e^{-cx}|} {|cx|^2(3-\gamma)_k|Q_{2k}(cx)|^2}. \tag{8} \]
An analogous estimate of the remainder term was obtained by A. A. Markov [3] only in the case of a real variable \(ax=z>0\) \((b=0,\ c=a)\).
After separating the real and imaginary parts of integral (7), formulas are obtained for computing the following integrals:
\[ \int_x^\infty t^{\gamma-1}e^{-at}\cos bt\,dt = \frac{AC+BD}{A^2+B^2}+Re\,[r_{2k}(cx)], \tag{9} \]
and also
\[ \int_x^\infty t^{\gamma-1}e^{-at}\sin bt\,dt = \frac{BC-AD}{A^2+B^2}-Im\,[r_{2k}(cx)], \tag{10} \]
where
\[ A=\sum_{n=0}^{k} C_k^n \frac{|c|^{\,n+2}x^{n+2}}{(3-\gamma)_n}\cos (n+2)\varphi, \tag{11} \]
\[ B=\sum_{n=0}^{k} C_k^n \frac{|c|^{\,n+2}x^{n+2}}{(3-\gamma)_n}\sin (n+2)\varphi, \tag{12} \]
\[ C=\frac{(1-\gamma)_2x^\gamma}{e^{ax}} \sum_{n=0}^{k+1}\frac{\cos(n\varphi-bx)}{|c|^{-n}x^{-n}} \sum_{m=0}^{k+1-n} \frac{C_k^m(-1)^{k+1-n-m}}{(2-\gamma+k-n-m)_{n+1}}, \tag{13} \]
\[ D=\frac{(1-\gamma)_2x^\gamma}{e^{ax}} \sum_{n=0}^{k+1}\frac{\sin(n\varphi-bx)}{|c|^{-n}x^{-n}} \sum_{m=0}^{k+1-n} \frac{C_k^m(-1)^{k+1-n-m}}{(2-\gamma+k-n-m)_{n+1}}. \tag{14} \]
The remainder terms of formulas (9) and (10) can be estimated on the basis of formula (8), since
\[ |Re\,[r_{2k}(cx)]|\le |r_{2k}(cx)| \]
and
\[ |Im\,[r_{2k}(cx)]|\le |r_{2k}(cx)|. \]
By successive application of the recurrence formulas ([4], p. 211) to the integrals (9), the following integrals are transformed:
\[ I_1=\int x^{\gamma+n-1}e^{-ax}\cos(bx-n\varphi)\,dx= \]
\[ =-e^{-ax}\sum_{k=0}^{n-1} \frac{(\gamma+n-k)_k}{|c|^{k+1}} x^{\gamma+n-1-k}\cos[(k+1-n)\varphi+bx]+ \]
\[ +\frac{(\gamma)_n}{|c|^n}\int x^{\gamma-1}e^{-ax}\cos bx\,dx, \tag{15} \]
\[ \int x^{\gamma-n-1}e^{-ax}\cos(bx+n\varphi)\,dx= \]
\[ =-e^{-ax}\sum_{k=0}^{n-1} \frac{(-1)^k|c|^k}{(-\gamma+n-k)_{k+1}} x^{\gamma-n+k}\cos[(n-k)\varphi+bx]+ \]
\[ +(-1)^n\frac{|c|^n}{(1-\gamma)_n} \int x^{\gamma-1}e^{-ax}\cos bx\,dx. \tag{16} \]
To prove relation (15), the derivative of its right-hand side is taken:
\[ \frac{dI_1}{dx} = e^{-ax}\sum_{k=0}^{n-1} \frac{(\gamma+n-k)_k}{|c|^{k+1}} x^{\gamma+n-1-k} \{a\cos[(k+1-n)\varphi+bx]+ \]
\[ +b\sin[(k+1-n)\varphi+bx]\}- \]
\[ - e^{-ax} \sum_{k=0}^{n-1} \frac{(\gamma+n-1-k)_{k+1}}{|c|^{k+1}} x^{\gamma+n-k-2}\cos\bigl[(k+1-n)\varphi+bx\bigr]+ \]
\[ +\frac{(\gamma)_n}{|c|^n}x^{\gamma-1}e^{-ax}\cos bx. \]
On the basis of (1), the expression in braces is equal to \(|c|\cos [(k-n)\varphi+bx]\); further, the last two terms cancel, and then the first term of the first sum is separated and the index \(m=k-1\) is introduced:
\[ \frac{d I_1}{dx} = x^{\gamma+n-1}e^{-ax}\cos(bx-n\varphi)+ \]
\[ +e^{-ax}\sum_{m=0}^{n-2} \frac{(\gamma+n-1-m)_{m+1}}{|c|^{m+1}} x^{\gamma+n-2-m}\cos\bigl[(m+1-n)\varphi+bx\bigr]- \]
\[ -e^{-ax}\sum_{k=0}^{n-2} \frac{(\gamma+n-1-k)_{k+1}}{|c|^{k+1}} x^{\gamma+n-2-k}\cos\bigl[(k+1-n)\varphi+bx\bigr] = \]
\[ = x^{\gamma+n-1}e^{-ax}\cos(bx-n\varphi). \]
Formula (15) is proved.
The relation (16) is proved analogously.
REFERENCES
-
V. L. Danilov, A. N. Ivanova et al. Mathematical Analysis (Functions, Limits, Series, Continued Fractions). Moscow, Fizmatgiz, 1961.
-
T. I. Stieltjes. Investigations on Continuous Fractions. Moscow, ONTI, 1936.
-
A. A. Markov. Selected Works on the Theory of Continued Fractions and the Theory of Functions Least Deviating from Zero. Moscow–Leningrad, Gostekhizdat, 1948.
-
I. S. Gradshteyn, I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Moscow, Fizmatgiz, 1962.