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absolute value becomes infinite branch circle BRANCH CURVE branch points CALIFORNIA PRESS BERKELEY CIRCLE AS BRANCH circle is approached const cosẫ cosh a cosh cosh cos n n cosh oz cosh rj deformed denotes the distance directional derivatives FINITELY LEAVED RIEMANN following theorem fundamental circle Hence lim infinitely leaved integral 3.2 integral sign Laplace's equation leaf LEAVED RIEMANN SPACE Legendre functions MULTIPLE VALUED HARMONIC multiple valued solution obtain oonst ordinary space path of integration plane point 17 pole 17o pole is approached readily seen real integral residue theorem RIEMANN SPACE SOMMERFELD rj cosh second term shown in figure SIEGFRIED F sinh rj sinh sinh tjo solution of Laplace's SPACE SOMMERFELD METHOD sufficiently large 17 system of toroidal Taylor series terms of Legendre toroidal coordinates underlying points UNIVERSITY OF CALIFORNIA VALUED HARMONIC FUNCTIONS vanishes at infinity Vcosh y/cosh