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Extra resources for Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications ~ Volume 106, Number 3 June, 2009 pp325-499
For now, interchanging the summation and integration yields ∞ p zk = lim k=0 p→∞ 0 = lim p→∞ dt e−t ∞ k=0 (zt)k = lim p→∞ k! 1 e−p(1−z) + . z−1 1−z p dt e−t (1−z) 0 (7) For z < 1, the first term in the last member of the above equation vanishes and the series yields a finite value of 1/(1 − z). Therefore, we see that the same value is obtained for the series when z < 1 as for when z lies in the circle of absolute convergence given by |z| < 1. According to the definition on p. 18 of Ref. , this means that the series is conditionally convergent for z < 1 and |z| > 1.
42) Generalizing the Reciprocal Logarithm Numbers 381 Equation (42) is not the only result involving an exponential integral of the gk (y).
Anal. Appl. 135, 501 (1988) 2. : Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Boston (1994) 3. : Exact and numerical traveling wave solutions of Whitham–Broer–Kaup equations. J. Comput. Appl. Math 167, 1339–1349 (2005) 4. , Ebrahimpour: Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys. Lett. A 368, 64–68 (2007) 5. : Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations.