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    A new method for the computation of the zeros of analytic functions
    Ioakimidis, Nikolaos; Anastasselou, Eleni; Ιωακειμίδης, Νικόλαος; Αναστασέλου, Ελένη
    Γενικό Τμήμα (Τεχνικές Αναφορές)
    A new method for the computation of the zeros of analytic functions (or the poles of meromorphic functions) inside or outside a closed contour C in the complex plane is proposed. This method is based on the Cauchy integral formula (in generalized forms) and leads to closed-form formulae for the zeros (or the poles) if they are no more than four. In general, for m zeros (or poles) these can be evaluated as the zeros of a polynomial of degree m. In all cases, complex contour integrals have to be evaluated numerically by using appropriate numerical integration rules. Several practical algorithms for the implementation of the method are proposed and the method of Abd-Elall, Delves and Reid is rederived by two different approaches as one of these algorithms. A numerical application to a transcendental equation appearing in the theory of neutron moderation is also made and numerical results of high accuracy are easily obtained.
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    A closed-form formula for the critical buckling load of a bar with one end fixed and the other pinned
    Anastasselou, Eleni; Ioakimidis, Nikolaos; Αναστασέλου, Ελένη; Ιωακειμίδης, Νικόλαος
    Γενικό Τμήμα (Τεχνικές Αναφορές)
    The classical problem of elastic buckling of a bar with one end fixed and the other pinned is reconsidered and a closed-form formula for the critical buckling load is derived. This is achieved through the closed-form solution (in terms of two regular integrals) of the transcendental equation tan u = u, to which this problem is reduced. The method of solution of this equation is too simple and based on a generalized form of the Cauchy theorem in complex analysis; yet the sought root of this equation does not contain complex quantities. Finally, numerical results verifying the validity of the derived formula are presented.
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    Formulae for the exponential, the hyperbolic and the trigonometric functions in terms of the logarithmic function
    Ioakimidis, Nikolaos; Anastasselou, Eleni; Ιωακειμίδης, Νικόλαος; Αναστασέλου, Ελένη
    Γενικό Τμήμα (Τεχνικές Αναφορές)
    A common definition of the exponential function is as the inverse function of the logarithmic function, which is defined as the definite integral of the rational function 1/t over the interval [1,x] with x > 0. The hyperbolic functions (hyperbolic sine, cosine, tangent, etc.) are next defined in terms of the exponential function. Here we derive an explicit real formula for the hyperbolic tangent function in terms of the logarithmic function, which is sufficient for the direct derivation of analogous formulae for the exponential function and the other hyperbolic functions. A similar formula for the trigonometric tangent function, which can be directly used for the derivation of analogous formulae for the other trigonometric functions, is also derived. The present results are based on a simple method for the derivation of closed-form formulae for the zeros of sectionally analytic functions.