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Mathematics · Advanced university

Holomorphic functions and Cauchy-Riemann equations

A derivative must have the same limit along every complex direction.

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Prerequisites

Complex arithmetic and partial derivatives.

Study notes

Complex differentiability

A derivative must have the same limit along every complex direction.

Cauchy-Riemann

For f=u+iv, differentiability implies u_x=v_y and u_y=−v_x when the partials exist.

Sufficient hypotheses

Continuous partial derivatives satisfying the equations on a neighborhood give holomorphicity there.

Not a point test

Equations at one isolated point alone do not prove holomorphicity throughout a neighborhood.

Worked examples

Verify the Cauchy-Riemann equations for f(z)=z².

Credit both equations and smoothness of the partials.

u=x²−y² and v=2xy. Then u_x=2x=v_y and u_y=−2y=−v_x.

For f(z)=z³, find the real derivative value f′(2).

f′(z)=3z², so f′(2)=12.

12

Check your understanding

  1. Is complex conjugation f(z)=conjugate(z) holomorphic on an open subset of C?

    1. It is a polynomial in z
    2. No
    3. Only because it is continuous
    4. Yes everywhere

  2. Use directional difference quotients to show conjugation is not complex differentiable at zero.

  3. What does a power series define inside its radius of convergence?

    1. A function with no derivative
    2. A necessarily constant function
    3. Only a function on the real axis
    4. A holomorphic function

  4. For f(z)=z² at z=i, write f′(i)=a i and find a.

Answers and marking guidance

Answers and marking notes
  1. No

    For u=x,v=−y, u_x=1 but v_y=−1, so the Cauchy-Riemann equations fail.

  2. Along real h, conjugate(h)/h=1; along imaginary h, the quotient is −1. The limits disagree.

    Credit two directions with distinct limiting quotients.

  3. A holomorphic function

    Complex power series can be differentiated termwise inside their convergence disk.

  4. 2

    f′(z)=2z, hence 2i.

Further reading and review references

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