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    Complex differentiability at a point without local holomorphy

    P-JHUFA09ANE

    Area
    Complex Analysis
    Topics
    Cauchy-Riemann Equations
    Status
    draft
    1. Let

    \[ f ( x + i y ) = x ^ { 3 } - 3 x y ^ { 2 } + i y ^ { 3 } . \]

    State whether each of the following is true or false and give proofs for your answers:

    1. the complex derivative \(f ^ { \prime } ( 0 )\) exists;

    2. f is holomorphic in a neighborhood of 0.

    Solution

    Part (a) is true, with \(f'(0)=0\). Part (b) is false.

    1. The difference quotient tends to zero at the origin.

    For \(z=x+iy\), one has \(f(0)=0\) and \(|x|,|y|\leq|z|\). The triangle inequality gives \[ |f(z)|\leq |x|^3+3|x||y|^2+|y|^3\leq5|z|^3. \] Thus for \(z\ne0\), \[ \left|\frac{f(z)-f(0)}z\right|\leq5|z|^2\longrightarrow0. \] This limit is independent of the direction of approach, so the complex derivative exists and equals zero.

    2. Every neighborhood contains a point where the complex derivative fails to exist.

    Writing \(f=u+iv\), its real partial derivatives are \[ u_x=3x^2-3y^2,\qquad u_y=-6xy,\qquad v_x=0,\qquad v_y=3y^2. \] Complex differentiability requires the Cauchy–Riemann equations \(u_x=v_y\) and \(u_y=-v_x\) [@SS03]. At a point \((x,0)\) with \(x\ne0\), the first equation would read \(3x^2=0\), which is false. Every neighborhood of zero contains such a point. Therefore no neighborhood of zero is a domain on which \(f\) is holomorphic, proving (b).

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