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Applications of Harmonic Functions — Howell & Matthews
Mathematics — Complex Analysis · poisson-integral-formula
Problem Show that the function ϕ ( x , y ) \phi(x,y) ϕ ( x , y ) ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) ϕ ( x , y ) = 1 π ∫ − ∞ ∞ U ( t ) [ ( ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 ) y ( x − t ) 2 + y 2 ] d t . \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)\phi(x,y) = \frac{1}{\pi}\int_{-\infty}^{\infty} U(t)\left[\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)\frac{y}{(x-t)^2+y^2}\right]dt. ( ∂ x 2 ∂ 2 + ∂ y 2 ∂ 2 ) ϕ ( x , y ) = π 1 ∫ − ∞ ∞ U ( t ) [ ( ∂ x 2 ∂ 2 + ∂ y 2 ∂ 2 ) ( x − t ) 2 + y 2 y ] d t . Browse the problem library