Calculus of Residua - Complex Functions Theory a-2 by Leif Mejlbro

By Leif Mejlbro

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6 that z + z0 z − z0 = r i(Θ−t) e R r i(Θ−t) = 1− e 1+ R 1+ (77) = R2 + r2 r 2 R +i r r 2 cos(Θ − t) −2 1+ R R 1− r sin(θ − t) R 2 r r cos(Θ − t) −2 R R 2 2Rr sin(Θ − t) R2 − r 2 +i 2 . − 2Rr cos(Θ − t) R + r2 − 2Rr cos(Θ − t) Finally, it follows from the Mean Value theorem for Harmonic Functions that 1 2π 2π 0 v(z) dt = 1 2π 2π v Reit dt = v(0), 0 and (71) follows from (76). If (77) is put into (76), we get (69) and (70) by splitting into the real and the imaginary parts.

A real interval to another analytic function g, then we change without any time delay f 39 43 Calculus of Residua Power Series to g over all of Ω. g. the impact of a sudden force applied to a system will evolve in time through the system and not immediately. Fortunately, it is possible to describe indirectly the causality (approximately) by analytic functions. However, this is not the right place to go further into this discussion. 4 states that there is at most one analytic function f : Ω → C, the restriction of which to an interval is a given function ϕ : I → C.

This follows immediately from the fact that every solution is uniquely determined by the n constants c0 , c1 , . . , cn−1 of (24). Practical procedures for solving a linear differential equations of analytical coefficients. In this subsection we shall in some examples demonstrate three standard procedures os solving a linear differential equation of analytical coefficients. These are 1) Inspection 2) Calculation of the germ f (n) (z0 ) n∈N0 3) Method of power series. Of these, inspection is the most difficult one.