Introducing Complex Numbers into Basic Growth Functions (3) : Symmetry Breakdown in Complex REpresentation of '0=(-1) + 1' int Definite Integral of Exponential Function with Base e Expanded into Infinite Series

This paper presents an alternative solution based on infinite series for the accurate and efficient evaluation of cable earth return impedances. This method uses Wedepohl and Wilcox’s transformation to decompose Pollaczek’s integral in a set of Bessel functions and a definite integral. The main feature of Bessel functions is that they are easy to compute in modern mathematical software tools such as Matlab. The main contributions of this paper are the approximation of the definite integral by an infinite series, since it does not have analytical solution; and its numerical solution by means of a recursive formula. The accuracy and efficiency of this recursive formula is compared against the numerical integration method for a broad range of frequencies and cable configurations. Finally, the proposed method is used as a subroutine for cable parameter calculation in the inverse Numerical Laplace Transform (NLT) to obtain accurate transient responses in the time domain.

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Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽

10.3390/math8091425 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1425

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Closed Form ◽

Special Function ◽

Special Functions ◽

Complex Numbers ◽

Definite Integral ◽

Hyperbolic Functions ◽

Closed Form Solutions ◽

Definite Integrals ◽

Arbitrary Complex ◽

Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

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On rearrangements of infinite series

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500033694 ◽

1958 ◽

Vol 3 (4) ◽

pp. 182-193 ◽

Cited By ~ 1

Author(s):

A. P. Robertson

Keyword(s):

Infinite Series ◽

Convergent Series ◽

Complex Numbers ◽

Famous Theorem

If a convergent series of real or complex numbers is rearranged, the resulting series may or may not converge. There are therefore two problems which naturally arise.(i) What is the condition on a given series for every rearrangement to converge?(ii) What is the condition on a given method of rearrangement for it to leave unaffected the convergence of every convergent series?The answer to (i) is well known; by a famous theorem of Riemann, the series must be absolutely convergent. The solution of (ii) is perhaps not so familiar, although it has been given by various authors, including R. Rado [7], F. W. Levi [6] and R. P. Agnew [2]. It is also given as an exercise by N. Bourbaki ([4], Chap. III, § 4, exs. 7 and 8).

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