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Series and Integral Representations
The Evolution of the Concept of Convergence
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absolute convergence analysis analytic continuation analytic function Applying asymptotic expansion asymptotic form asymptotic formula asymptotic series boundary calculate called Cauchy problem change of variable coefficients complex plane condition Consider constant constructed contour of integration curve defined definition denote diagonal disc of convergence divergent series domain eigenvalues equal Euler example exponentially find the asymptotic follows formal series Fourier integrals Fourier transform function f(x gamma-function improper integral infinite integral representation integral transforms integrand inversion formula Laplace transform Laplace's method leading term lemma level lines linear many-dimensional mathematical matrix Mellin transform nondegenerate notation obtain operator ordinary differential equations parameter point x0 poles polynomial power series real axis reduces result Riemann saddle contour saddle point satisfies second order sector simple turning point singular points space stationary point steepest descent Stokes lines takes the form tends to zero theorem theory Weierstrass