We know that log∣f∣=21log(ffˉ). If we write f=u+iv, where u,v are real functions, we find log(ffˉ)=log(u2+v2).
Therefore, we have
(∂x2∂2+∂y2∂2)log∣f∣=21(∂x2∂2+∂y2∂2)log(u2+v2) and let us calculate the derivatives :
21∂x2∂2log(u2+v2)=∂x(u2+v2u∂xu+v∂xv) taking the second derivative gives us a long expression :
(u2+v2)(u∂xxu+v∂xxv+(∂xu)2+(∂xv)2)−−2u2(∂xu)2−2v2(∂xv)2−4uv∂xu∂xv all divided by (u2+v2)2
Developping this expression gives us
(u2+v2)2(u2+v2)(u∂xxu+v∂xxv)+(v∂xu−u∂xv)2−(u∂xu+v∂xv)2
Calculating the second derivative with respect to y gives us a similar expression :
(u2+v2)2(u2+v2)(u∂yyu+v∂yyv)+(v∂yu−u∂yv)2−(u∂yu+v∂yv)2
Now let us use the Cauchy-Riemann equations (as f is analytic, u,v should satisfy them) :
{∂xu=∂yv∂yu=−∂xv and therefore (∂x2∂2+∂y2∂2)u=(∂x2∂2+∂y2∂2)v=0
In addition, we have (u∂yu+v∂yv)2=(v∂xu−u∂xv)2 and (u∂xu+v∂xv)2=(u∂yv−v∂yu)2 by Cauchy-Riemann equations. Therefore, the sum
(∂x2∂2+∂y2∂2)log∣f∣=21(∂x2∂2+∂y2∂2)log(u2+v2)=0
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