Question 351313
Harmonic means that a function satisfies the Laplace equation, {{{fxx+fyy=0}}}.
({{{fx}}} is the partial derivative of {{{f}}} with respect to {{{x}}}, {{{fxx}}} is partial derivative of {{{fx}}} with respect to {{{x}}})
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{{{f(x,y)=e^x*cos(y)}}}
{{{fx=e^x*cos(y)}}}
{{{fxx=e^(x)*cos(y)}}}
{{{fy=e^x(-sin(y))}}}
{{{fyy=-e^(x)*sin(y)}}}
{{{fxx+fyy=e^x*cos(y)-e^(x)*cos(y)=0}}}, so the function is harmonic.
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{{{f(x,y)=ln(x^2+y^2)}}}
{{{fx=2x/log(x^2+y^2)}}}
{{{fxx=2/(x^2+y^2)-4x^2/(x^2+y^2)^2}}}
{{{fy=2y/log(x^2+y^2)}}}
{{{fyy=2/(x^2+y^2)-4y^2/(x^2+y^2)^2}}}
{{{fxx+fyy=4/(x^2+y^2)-4(x^2+y^2)/(x^2+y^2)^2}}}
{{{fxx+fyy=4/(x^2+y^2)-4/(x^2+y^2)}}}
{{{fxx+fyy=0}}}
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I leave the last one for you to finish.
Follow the same procedure.