Suppose f(z) = u(x, y) is a real valued function on an open set U which we are viewing as a complex-valued function (in other words, we take v(x, y) = 0.) a) Show that if zo = xo + iyo is such that f'(zo) exists, then one has dru(xo, yo) = dyu(xo, yo) = 0 {z: z0}. Show that f'(zo) does not exist at any b) Let f(z) = |2| on U 30 € U. =

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A: (.)  Leibniz rule :      Given ,(A).        where   and   .

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Q: f (x, y) ~ bo + b1,0(x − 1)+b0,1(y−2)+ b2,0 (x - 1)² + b₁,1(x − 1)(y-2) + bo,2(y − 2)² +…...

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Q: Show that f(x) = 1/(1+x) is analytic on (-1, ∞).

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Q: 1.16. (a) If f and g are differentiable scalar functions, show that V(fg)=fVg+gVf.

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