APPLICATION OF THE THEORY OF CONTINUED FRACTIONS TO THE EVALUATION OF CERTAIN TYPES OF INTEGRALS

V. E. KORNILOV

(Presented by the Department of Higher Mathematics)

In the paper the parameters \(0<\gamma<1\), \(c=a+bi\), \(|\arg c|\le \dfrac{\pi}{2}\), \(\varphi=\operatorname{arctg}\dfrac{b}{a}\), and the independent variable \(x>0\) are used.

In the paper [1] the corresponding continued fraction for the series

\[ \frac{1}{cx}F\left(3-\gamma,\,-\frac{1}{cx}\right) \]

was used to evaluate integrals of the form

\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt. \]

In this paper, for the evaluation of the integrals \(I\), the corresponding continued fraction is applied to the series

\[ \frac{1}{cx}F\left(2-\gamma,\,-\frac{1}{cx}\right) \quad\text{and} \]

\[ \frac{1}{cx}F\left(1-\gamma,\,-\frac{1}{cx}\right) = \sum_{k=0}^{\infty}(-1)^k\frac{(1-\gamma)_k}{(cx)^{k+1}}, \qquad |cx|>1, \]

where

\[ (a)_k=a(a+1)\cdots(a+k-1)=\frac{\Gamma(a+k)}{\Gamma(a)}. \]

1. Analogously to the derivation in paper [1], the integral \(I\) may be represented by a suitable fraction, and the remainder term in the form of a functional series:

\[ I=\int_x^\infty t^{\gamma-1}e^{-ct}\,dt = x^\gamma e^{-cx}\frac{P_{2k}(cx)}{Q_{2k}(cx)} +r_{2k}(cx)= \]

\[ = \frac{ x^\gamma e^{-cx} \displaystyle\sum_{n=0}^{k-1}(cx)^n \displaystyle\sum_{m=0}^{k-1-n} \frac{C_k^m(-1)^{k-1-n-m}}{(k-\gamma-n-m)_{n+1}} }{ \displaystyle\sum_{n=0}^{k} C_k^n\frac{(cx)^n}{(1-\gamma)_n} } +r_{2k}(cx) = \frac{C_{2k}+D_{2k}i}{A_{2k}+B_{2k}i} +r_{2k}(cx), \qquad k=1,2,\ldots . \tag{1} \]

where

\[ r_{2k}(cx)=x^{\gamma}e^{-cx}\sum_{n=k}^{\infty} \frac{n!}{(1-\gamma)_{n+1}Q_{2n}(cx)Q_{2n+2}(cx)} . \tag{2} \]

After separating the real and imaginary parts of integral (1), one can obtain a sequence of approximations that are used to compute the following integrals:

\[ I_1=\int_x^\infty t^{\gamma-1}e^{-at}\cos bt\,dt= \]

\[ = \frac{ x^\gamma e^{-ax} \displaystyle\sum_{n=0}^{2k-1}|c|^n x^n \displaystyle\sum_{m=N}^{M} C_k^m \frac{1}{(1-\gamma)_m} B_{k-1}^{\,n-m}\cos [bx-(n-2m)\varphi] }{ \displaystyle\sum_{n=0}^{k} C_k^n \frac{(1-\gamma+k)_n}{(1-\gamma)_n(1-\gamma)_{2n}} |cx|^{2n} \displaystyle\sum_{m=0}^{k-n} C_{k-n}^{m} \frac{(2ax)^m}{(1+2n-\gamma)_m} } + \]

\[ +\,\operatorname{Re}[r_{2k}(cx)] = \frac{A_{2k}C_{2k}+B_{2k}D_{2k}}{A_{2k}^{2}+B_{2k}^{2}} +\operatorname{Re}[r_{2k}(cx)] , \tag{3} \]

and also

\[ I_2=\int_x^\infty t^{\gamma-1}e^{-at}\sin bt\,dt= \]

\[ = \frac{ x^\gamma e^{-ax} \displaystyle\sum_{n=0}^{2k-1}|c|^n x^n \displaystyle\sum_{m=N}^{M} C_k^m \frac{1}{(1-\gamma)_m} B_{k-1}^{\,n-m}\sin [bx-(n-2m)\varphi] }{ |Q_{2k}(cx)|^2 } -\operatorname{Im}[r_{2k}(cx)] , \tag{4} \]

where

\[ B_{k-1}^{\,n-m} = \sum_{p=0}^{k-1-n+m} C_k^p \frac{(-1)^{k-1-n+m-p}}{(k-\gamma-n+m-p)_{n-m+1}} ; \tag{5} \]

\[ N= \begin{cases} 0, & \text{for } n<k-\dfrac12,\\[4pt] n-k+1, & \text{for } n>k-\dfrac12; \end{cases} \tag{6} \]

\[ M= \begin{cases} n, & \text{for } n<k+\dfrac12,\\[4pt] k, & \text{for } n>k+\dfrac12. \end{cases} \tag{7} \]

Let us also write the approximations with odd indices for the integral \(I_1\):

\[ I_1=\int_x^\infty t^{\gamma-1}e^{-at}\cos bt\,dt = \operatorname{Re}\left[ \frac{x^\gamma e^{-cx}P_{2k+1}(cx)} {cx\,Q_{2k+1}(cx)} \right] +\operatorname{Re}[r_{2k+1}(cx)] = \]

\[ = \frac{ x^\gamma e^{-ax} \displaystyle\sum_{n=0}^{2k}|c|^n x^n \displaystyle\sum_{m=N}^{M} C_k^m \frac{(1-\gamma)}{(2-\gamma)_m} B_k^{\,n-m} \cos [bx-(n-1-2m)\varphi] }{ |cx| \displaystyle\sum_{n=0}^{2k} C_k^n \frac{(2-\gamma+k)_n}{(2-\gamma)_n(2-\gamma)_{2n}} |cx|^{2n} \displaystyle\sum_{m=0}^{k-n} C_{k-n}^m \frac{(2ax)^m}{(2+2n-\gamma)_m} } + \]

\[ +\operatorname{Re}[r_{2k+1}(cx)] . \tag{8} \]

where \(M\) is computed by formula (7),

\[ N= \begin{cases} 0, & \text{for } n<k+\dfrac{1}{2},\\[4pt] n-k, & \text{for } n>k+\dfrac{1}{2}. \end{cases} \]

\[ B_k^{\,n-m}=\sum_{p=0}^{k-n+m} C_k^p \frac{(-1)^{k-n+m-p}}{(1-\gamma+k-n+m-p)_{n-m+1}} . \tag{9} \]

Moreover, in view of [2], pp. 23, 34,

\[ r_{2k+1}(cx)= -\frac{x^\gamma e^{-cx}}{(cx)} \sum_{n=k}^{\infty} \frac{(1-\gamma)\,n!}{(2-\gamma)_{n+1}Q_{2n+1}(cx)Q_{2n+3}(cx)} . \tag{10} \]

In formula (8) the denominator is the following:

\[ Q_{2k+1}(cx)=\sum_{n=0}^{k} C_k^n \frac{(cx)^n}{(2-\gamma)_n}. \tag{11} \]

1. To compute the numerator of equality (3), equality (5) and the separation into the real and imaginary parts of the numerator and denominator of equality (1) are taken into account; therefore we obtain:

\[ A_{2k}C_{2k}+B_{2k}D_{2k}= \]

\[ = x^\gamma e^{-ax} \sum_{n=0}^{k-1} |c|^n x^n B_{k-1}^n \left\{ \sum_{m=0}^{k} C_k^m \frac{|c|^m x^m}{(1-\gamma)_m} \cos\bigl[(n-m)\varphi-bx\bigr] \right\} = \]

\[ = \frac{x^\gamma}{e^{ax}} \left\{ B_{k-1}^0 \left[ \cos(-bx)+ \frac{|c|x}{1-\gamma}\cos(-\varphi-bx)+\cdots \right.\right. \]

\[ \left. \left. \cdots+ \frac{|c|^k x^k}{(1-\gamma)_k}\cos(-k\varphi-bx) \right] + B_{k-1}^1 |c|x \left[ \cos(\varphi-bx)+\cdots \right. \right. \]

\[ \left. \left. \cdots+ \frac{|c|^k x^k}{(1-\gamma)_k} \cos(\varphi-k\varphi-bx) \right] +\cdots \right. \]

\[ \left. \cdots+ B_{k-1}^{k-1}|c|^{k-1}x^{k-1} \left[ \cos(k\varphi-\varphi-bx) + \frac{|c|x}{1-\gamma}\cos(k\varphi-2\varphi-bx) +\cdots \right. \right. \]

\[ \left. \left. \cdots+ \frac{|c|^k x^k}{(1-\gamma)_k} \cos(-\varphi-bx) \right] \right\} = \]

\[ = x^\gamma e^{-ax} \sum_{n=0}^{2k-1} |c|^n x^n \sum_{m=N}^{M} C_k^m \frac{1}{(1-\gamma)_m} B_{k-1}^{\,n-m} \cos\bigl[bx-(n-2m)\varphi\bigr]. \]

Here \(N\), \(M\), and \(B_{k-1}^{\,n-m}\) take the values and are computed according to equalities (6), (7), and (15).

The numerators of equalities (4) and (8) are transformed analogously.

1. Taking into account the formula for the Chebyshev polynomials [3]

\[ T_n(x)=\cos(n\arccos x)= \frac{1}{2} \sum_{m=0}^{\left[\frac{n}{2}\right]} (-1)^m \frac{n}{n-m} C_{n-m}^m(2x)^{n-2m}, \]

where \([\nu]\) denotes the greatest integer not exceeding \(\nu\); next we compute the square of the modulus of the denominator of equality (1) in the following way:

\[ \begin{aligned} \left|\sum_{n=0}^{\kappa} C_\kappa^n \frac{(cx)^n}{(1-\gamma)_n}\right|^2 &= \left|\sum_{n=0}^{\kappa}\sigma_n c^n x^n\right|^2 =\sum_{n=0}^{\kappa}\sigma_n^2 |cx|^{2n}+ \\ &\quad +2\sum_{n=0}^{\kappa}\sum_{m=0}^{\kappa-n} \sigma_{n-l}\sigma_{\kappa-m+l} \cos\left[(\kappa-m-n+2l)\arccos\frac{a}{|c|}\right] |cx|^{\kappa+n-m}= \\ &= \sum_{n=0}^{\kappa}|cx|^{2n}\sum_{m=0}^{\kappa-n}(2ax)^{\kappa-n-m} \sum_{l=0}^{m,n}(-1)^l \frac{\kappa-m-n+2l}{\kappa-m-n+l} C_{\kappa-m-n+l}^{\,l}\sigma_{n-l}\sigma_{\kappa-m+l}= \\ &= \sum_{n=0}^{\kappa}|cx|^{2n}\sum_{m=0}^{\kappa-n} C_\kappa^n C_{\kappa-n}^{m} \frac{(1+\kappa-\gamma)_n}{(1-\gamma)_{\kappa+n-m}(1-\gamma)_n} (2ax)^{\kappa-n-m}= \\ &= \sum_{n=0}^{\kappa} \frac{(1-\gamma+\kappa)_n}{(1-\gamma)_n(1-\gamma)_{2n}} |cx|^{2n} \sum_{m=0}^{\kappa-n} C_{\kappa-n}^{m} \frac{(2ax)^m}{(1+2n-\gamma)_m}. \end{aligned} \]

In the penultimate equality the identity was applied

\[ \begin{aligned} \frac{C_\kappa^n C_\kappa^m}{(1-\gamma)_n(1-\gamma)_{\kappa-m}}\lambda_{\kappa,m}^n &= \sum_{l=0}^{m,n}(-1)^l \frac{\kappa-n-m+2l}{\kappa-n-m+l} C_{\kappa-n-m+l}^{\,l} C_\kappa^{\,n-l} C_\kappa^{\,\kappa+l-m} \times \frac{1}{(1-\gamma)_{n-l}}\times \\ &\quad \times \frac{1}{(1-\gamma)_{\kappa+l-m}} = C_\kappa^n C_{\kappa-n}^{m} \frac{(1-\gamma+\kappa)_n}{(1-\gamma)_{\kappa+n-m}(1-\gamma)_n}, \end{aligned} \tag{12} \]

where \([\nu]\) denotes the greatest integer not exceeding \(\nu\); further,

1. Let us prove identity (12) in the case \(m\le n\).
   After simplifying equality (12) and solving with respect to \(\lambda_{\kappa,m}^n\), we obtain:

\[ \begin{aligned} \lambda_{\kappa,m}^n &=1+\sum_{l=1}^{m} C_m^l(-1)^l \times \\ &\quad \times \frac{(\kappa-n-m+2l)(1+\kappa-n-m)_{l-1}(-n)_l(\gamma-n)_l} {(1+\kappa-n)_l(1+\kappa-m)_l(1-\gamma+\kappa-m)_l} = \\ &= \frac{(1+\kappa-n-m)_m(1-\gamma+\kappa+n-m)_m} {(1+\kappa-m)_m(1-\gamma+\kappa-m)_m}. \end{aligned} \tag{13} \]

By direct verification we are convinced that identity (13) is valid for \(m=1;\ 2\); suppose that it is valid when the upper limit of the sum is equal to \(m-1\). Then, when the upper limit is \(m\), we transform the sum on the left-hand side of equality (13) in the following way:

\[ \lambda_{\kappa,m}^n-1 = (\lambda_{\kappa,m}^n-1) \frac{(\kappa-n)C_m^l}{(\kappa-n)C_m^l} = \]

\[ = (\lambda_{\kappa,m}^n-1) \frac{ \kappa C_{m-1}^l-nC_{m-1}^l+lC_{m-1}^l+\kappa C_{m-1}^{l-1}-nC_{m-1}^{l-1}-lC_{m-1}^l } {(\kappa-n)C_m^l}, \]

Next we obtain

\[ \begin{aligned} \lambda_{\kappa,m}^{n} &=1+\sum_{l=1}^{m-1} C_{m-1}^{l}(-1)^l \times \\ &\quad \times \frac{(-n)_l(\kappa-n-m+2l)(1+\kappa-n-m)_{l-1}(\gamma-n)_l} {(1+\kappa-m)_l(\kappa-n)_l(1-\gamma+\kappa-m)_l} + \\ &\quad +\sum_{l=1}^{m-1} C_{m-1}^{l-1}(-1)^l \frac{(-n)_l(\kappa-n-m+2l)(1+\kappa-n-m)_l(\gamma-n)_l} {(1+\kappa-m)_l(\kappa-n)_{l+1}(1-\gamma+\kappa-m)_l} = \\ &=\lambda_{\kappa-1,m-1}^{n} -\left[ 1+\sum_{l=1}^{m-1} C_{m-1}^{l}(-1)^l \times \right.\\ &\quad \left. \times \frac{(1-n)_l(\kappa-n-m+2+2l)(\kappa-n-m+3)_{l-1}(\gamma-n+1)_l} {(2+\kappa-m)_l(2+\kappa-n)_l(2-\gamma+\kappa-m)_l} \right]\times \\ &\quad \times \frac{n(\kappa-n-m+1)_2(n-\gamma)} {(1+\kappa-m)(\kappa-n)_2(1-\gamma+\kappa-m)} =\lambda_{\kappa-1,m-1}^{n}- \\ &\quad -\lambda_{\kappa,m-1}^{\,n-1} \frac{n(\kappa-n-m+1)_2(n-\gamma)} {(1+\kappa-m)(\kappa-n)_2(1-\gamma+\kappa-m)} = \\ &= \frac{(1+\kappa-m-n)_{m-1}(1-\gamma+\kappa-m+n)_{m-1}} {(1+\kappa-m)_{m-1}(1-\gamma+\kappa-m)_{m-1}} - \\ &\quad - \frac{(1+\kappa-m-n)_{m-1}(1-\gamma+\kappa-m+n)_{m-1}n(n-\gamma)} {(1+\kappa-m)_m(1-\gamma+\kappa-m)_m}; \end{aligned} \]

that is,

\[ \lambda_{\kappa,m}^{n}= \frac{(1+\kappa-n-m)_m(1-\gamma+\kappa-m+n)_m} {(1+\kappa-m)_m(1-\gamma+\kappa-m)_m}, \]

thus identities (13) and (12) for \(m\le n\) are proved; identity (12) in the case \(m>n\) is proved analogously (the upper limit of the sum in the left-hand side of equality (12) is then taken equal to \(n\)).

In view of the foregoing, the values of the denominators of equalities (3), (4) have been proved, namely

\[ \begin{aligned} A_{2k}^{2}+B_{2k}^{2} &=\lvert Q_{2k}(cx)\rvert^{2}= \\ &=\sum_{n=0}^{k} C_k^n \frac{(1-\gamma+k)_n}{(1-\gamma)_n(1-\gamma)_{2n}} \lvert cx\rvert^{2n} \sum_{m=0}^{k-n} C_{k-n}^{m} \frac{(2ax)^m}{(1+2n-\gamma)_m}, \end{aligned} \tag{14} \]

the denominator of equality (8) is obtained from equality (14) by increasing the parameter \(1-\gamma\) by one.

5. Theorem. Along with the inequality

\[ \lvert Q_{2k}(cx)\rvert^{2}-Q_{2k}(2ax)<\lvert Q_{2k}(cx)\rvert^{2}, \tag{15} \]

the inequality

\[ \frac{1}{(k+1)(k+2-\gamma)} \left[\lvert Q_{2k+2}(cx)\rvert^{2}-Q_{2k+2}(2ax)\right]> \]

\[ >\frac{1}{k(k+1-\gamma)} \left[\lvert Q_{2k}(cx)\rvert^{2}-Q_{2k}(2ax)\right]. \]

Proof. Let us replace the squares of the moduli of the polynomials according to formula (14); we obtain:

\[ \begin{gathered} \frac{1}{(\kappa+1)(\kappa+2-\gamma)} \sum_{n=1}^{\kappa+1} C_{\kappa+1}^{n} \frac{(2-\gamma+\kappa)_n}{(1-\gamma)_n(1-\gamma)_{2n}} |cx|^{2n} \sum_{m=0}^{\kappa+1-n} C_{\kappa+1-n}^{m} \frac{(2ax)^m}{(1+2n-\gamma)_m} \\ - \frac{1}{\kappa(\kappa+1-\gamma)} \sum_{n=1}^{\kappa} C_{\kappa}^{n} \frac{(1-\gamma+\kappa)_n}{(1-\gamma)_n(1-\gamma)_{2n}} |cx|^{2n} \\ {}\times \sum_{m=0}^{\kappa-n} C_{\kappa-n}^{m} \frac{(2ax)^m}{(1+2n-\gamma)_m} > \\ > \sum_{n=1}^{\kappa} C_{\kappa}^{n} \frac{(2-\gamma+\kappa)_{n-1}}{(1-\gamma)_n(1-\gamma)_{2n}} |cx|^{2n} \sum_{m=0}^{\kappa-n} C_{\kappa-n}^{m} \frac{(2ax)^m}{(1+2n-\gamma)_m} \\ {}\times \frac{(2n+m-2)\kappa+(m+n-1)(2-\gamma)} {\kappa(\kappa+2-\gamma)(1+\kappa-n-m)} >0 . \end{gathered} \]

Inequality (16) is thereby proved.

1. Let us estimate the remainder term (2) of the rational approximations (1) in modulus, taking inequalities (15) and (16) into account:

\[ |r_{2\kappa}(cx)|< x^{\gamma}e^{-ax}\sum_{n=\kappa}^{\infty} \]

\[ {}\times \frac{n!} {(1-\gamma)_{n+1} \sqrt{|Q_{2n}(cx)|^2-Q_{2n}(2ax)} \cdot \sqrt{|Q_{2n+2}(cx)|^2-Q_{2n+2}(2ax)}} < \]

\[ < \frac{x^{\gamma}e^{-ax}\,\kappa!\sqrt{(\kappa)_2(\kappa+1-\gamma)_2}} {(1-\gamma)_{\kappa+1} \sqrt{|Q_{2\kappa}(cx)|^2-Q_{2\kappa}(2ax)} \cdot \sqrt{|Q_{2\kappa+2}(cx)|^2-Q_{2\kappa+2}(2ax)}} \]

\[ {}\times \left[ \sum_{n=0}^{\infty} \frac{(\kappa+1)_n} {(\kappa+2-\gamma)_n\sqrt{(\kappa+n)_2(\kappa+n+1-\gamma)_2}} \right] < \]

\[ < \frac{x^{\gamma}e^{-ax}(\kappa+1)!(\kappa+1-\gamma)} {(1-\gamma)_{\kappa+1}\left[|Q_{2\kappa}(cx)|^2-Q_{2\kappa}(2ax)\right]} \sum_{n=0}^{\infty}\frac{1}{(\kappa+n)_2}, \]

that is,

\[ |r_{2\kappa}(cx)|< \frac{x^{\gamma}e^{-ax}(\kappa+1)!} {\kappa(1-\gamma)_{\kappa}\left[|Q_{2\kappa}(cx)|^2-Q_{2\kappa}(2ax)\right]} . \tag{17} \]

The remainder term (10) is estimated analogously, namely:

\[ |r_{2\kappa+1}(cx)|< \frac{(1-\gamma)x^{\gamma}e^{-ax}(\kappa+1)!} {\kappa(2-\gamma)_{\kappa}|cx|\left[|Q_{2\kappa+1}(cx)|^2-Q_{2\kappa+1}(2ax)\right]} . \tag{18} \]

If we take into account the value of the following integral [4, p. 205]:

\[ \int_{0}^{\infty} t^{\gamma-1}e^{-at}\cos(bt)\,dt = \frac{\cos(\gamma\varphi)}{|c|^{\gamma}}\Gamma(\gamma), \qquad 0<\gamma<1, \tag{19} \]

then, in view of formulas (19) and (3), we obtain

\[ \int_{0}^{x} t^{\gamma-1}e^{-at}\cos(bt)\,dt = \frac{\cos(\gamma\varphi)}{|c|^{\gamma}}\Gamma(\gamma)-I_1, \qquad 0<\gamma<1, \tag{20} \]

where \(I_1\) is computed according to formula (3).

References

1. V. E. Kornilov. Application of continued fractions to the evaluation of certain types of integrals. Izv. TPI, vol. 131, pp. 26–30, 1965.

2. T. I. Stieltjes. Investigations on continued fractions. Moscow, ONTI, 1936.

3. V. L. Danilov, A. N. Ivanova et al. Mathematical Analysis (Functions, Limits, Series, Continued Fractions). Moscow, Fizmatgiz, 1961.

4. I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Moscow—Leningrad, Gostekhizdat, 1948.