Asymptotic Description of the Cusps of a Hydromagnetic Figure of Equilibrium
Asymptotic Description of the Cusps of a Hydromagnetic Figure of Equilibrium
Robert E Dowd
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We note that t is really the branch point of Vg(t) - x which we previously labelled t*. To avoid looping about the branch point t we must choose the path of t so that it always passes to the right of t . Then g(t) will keep to the right of z . This is illustrated in figure 8. For z = z we cause K o the path of t to follow a half circle to the right of t so that g(t) will follow a little detour to the right of z, . For z = z, a much greater deformation of the path of t is required to keep to the... right of t . This keeps g(t) to the right of z, . Our interest is primarily in z_ = x. Here we lead the path of in- o tegration out to and around t='= and back uncfer the axis to x. This causes g(t) to move out to and around z and back under the axis to g(z ). We would like to be free to use any path of integration in the upper half plane, especially the real negative x-axis itself. As we have just shown this n ~2~ cannot be done for g(z) = z - a iz . If the branch point t* were placed below the real negative x-axis, however, we could choose the portion of this axis from to x as our path of integration for I.
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