Geodesic
Integrals of the Loewner equation with exponential driving function
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 4, pp. 98-108

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We consider the qualitative local behavior of trajectories for the ordinary Loewner differential equation with a driving function which is inverse to the exponential function of an integer power. All the singular points and the corresponding singular solutions are described. It is shown that this driving function generates solutions to the Loewner equation which map conformally a half-plane slit along a smooth curve onto the upper half-plane. The asymptotical correspondence between harmonic measures of two slit sides is derived. 
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     author = {D. V. Prokhorov and K. A. Samsonova},
     title = {Integrals of the {Loewner} equation with exponential driving function},
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D. V. Prokhorov; K. A. Samsonova. Integrals of the Loewner equation with exponential driving function. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 4, pp. 98-108. http://geodesic.mathdoc.fr/item/ISU_2013_13_4_a16/