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Aimed toward the neighborhood of mathematicians engaged on usual and partial differential equations, distinction equations, and sensible equations, this ebook comprises chosen papers in line with the shows on the foreign convention on Differential & distinction Equations and purposes (ICDDEA) 2015, devoted to the reminiscence of Professor Georg promote.

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E x a m p l e 1 Since cos ζ is entire, has antiderivative sin z, and ^ is simply connected, we have cos ζ dz = sin ζ = 2 sin i = 2i sinh(l). and along any pwd closed curve γ cos ζ dz = 0. E x a m p l e 2 The function l/z, analytic in ζΦΟ, has antiderivative log ζ. In this case care must be taken to specify the domain G. Suppose G is given by I arg ζ | < π, then for any arc joining — / to / in G ' dz ^ — = Log ζ -i ζ = πι. On the other hand, if G' is the domain given by 0 < arg ζ < 2π, we have dz — = log ζ Finally, letting y: ζ{θ) = e'\ π/2 < Ö < 3π/2, we obtain dz |z| = l 3π ζ -i r in Ζ = r dz -π/2<ο<π/2, and γ': ζ(θ) = e'", r dz — + ^ r —ζ = πι Jy ζ since y lies in G and y' lies in G\ Example 3 | r sin(x") ^ π dx = - .

Sin ζ z^ + 1 (b) dz. (d) ' cos ζ dz. sin ζ ζ — ζ ί/ζ. 2 . 4 T H E C A U C YH I N T E G R A L 53 F O R M U L A 3. Let y: z(t) = 2e'^ 4- 1, 0 < ί < 2π. Evaluate the following integrals: (a) dz, cos ζ (b) (z-1) 'y (c) sin ζ sin ζ dz, id) dz, 2 dz. 4. Let / ( ζ ) be analytic in |z - C| < ^ . Prove Gauss's Mean-Value /(() = -L r'7(C 2π -^0 + re'') de, Theorem 0

Find all the zeros of sinh ζ and cosh z. 10. Show that the function vv = sin ζ maps (a) (b) (c) the strip | A : | < π/2 onto ^ - {z: j = 0, \x\ > 1}, the semiinfinite strip \x\ < n/l, y>0 onto the upper half plane, the semiinfinite strip 0 < χ < π/2, y >0 onto the first quadrant, by indicating what happens to horizontal and vertical line segments under the transformation w = sin ζ = sin χ cosh y + i cos χ sinh y. NOTES Section LI Formulas relating ζ to Ζ in the stereographic projection are easy to compute: [A, pp.