Integrals of Legendre polynomials over half range and their relation to the electrostatic potential in hemispherical geometry

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Spherical symmetry is often encountered in many problems in mathematics and physical sciences. On the other hand, cases that involve hemispherical geometry lead to much less studied mathematical situations. In this work, we consider a case study that arises from potential theory where absence of a full spherical symmetry gives rise to expressions involving Legendre polynomials constrained in an interval that is different from the usual range in which they are orthogonal. We use exact mathematical results for the integrals of Legendre polynomials of arbitrary order over half range to obtain analytic expressions for the electrostatic potential created by a solid hemisphere with uniform volume charge density.

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