Question #15754

1. Find a harmonic conjugate of the function u(x,y)=cos x cosh y.
2. Let u(x,y)= ln (x^2+y^2) for (x,y) ∈R^2 \ {(0,0)}. Show that, although u is harmonic, there exists no f analytic on C \ {(0,0)} such that u= Re f. [ you must show u is indeed harmonic on the specified domain first]
3.For all a,b,c are complex numbers , we have :
i. a^b*a^c=a^b+c
ii. a^c*b^c=(ab)^c
iii. If a=b, then a^c= b^c

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