Calculus Small Changes, Big Ideas

Section 17.4 The Cauchy-Riemann equations

• Derivative of complex function:

  \begin{equation*} f'(z)=\limit{h\to 0}\frac{f(z+h)-f(z)}{h} \end{equation*}

  Note that \(h\) is complex here! Alternatively:

  \begin{equation*} f'(z_{0})=\limit{z\to z_0}\frac{f(z)-f(z_{0})}{z-z_{0}} \end{equation*}

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• Limits are defined exactly as you think they are based on multivariable calculus:

  \begin{equation*} \limit{z\to z_0}f(z)=L \end{equation*}

  means that \(f(z)\) approaches \(L\) as closely as desired when \(z\) approaches \(z_{0}\) along any path.

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• Some functions aren’t differentiable. Great example is \(f(z)=\overline{z}\text{:}\)

  Aside

  \begin{equation*} \limit{h\to 0}\frac{\overline{z+h}-\overline{z}}{h}=\limit{h\to 0}\frac{\ \ \overline{h}\ \ }{h} \end{equation*}

  If \(h\to 0\) along the real line, this quotient approaches \(1\text{,}\) but if it approaches over the imaginary line, the quotient approaches \(-1\text{.}\) Therefore the limit does not exist, and hence the complex conjugate is not differentiable.... anywhere!

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• If a function \(f\) is differentiable at \(z_{0}\) and at every point in some neighborhood of \(z_{0}\text{,}\) we say \(f\) is holomorphic at \(z_{0}\text{.}\)

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• If a function \(f(z)=u(x,y)+\ii v(x,y)\) is differentiable at a point, it must satisfy the Cauchy-Riemann equations:

  \begin{equation*} \pdv{u}{x}=\pdv{v}{x}\qquad \pdv{u}{y}=-\pdv{v}{y} \end{equation*}

  Furthermore, if \(u\) and \(v\) are continuous and have continuous first partial derivatives in a domain \(D\text{,}\) then satisfying the Cauchy-Riemann equations at all points of \(D\) implies that \(f\) is holomorphic in \(D\text{.}\)

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• All the “nice” functions are holomorphic in their domains: polynomial, rational, exponential, trigonometric, logarithmic, inverse trigonometric. (Excluding branch cuts.) This in fact means that all our original derivative formulas from calculus apply!

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• If \(f\) is holomorphic in a domain \(D\text{,}\) then \(f\) is constant if \(|f|\) is constant or if \(f'=0\text{.}\)

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• A function that is holomorphic at every point in the complex plane is called entire.

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• Linear approximation:

  \begin{equation*} f(z)=f(z_{0})+f'(z_{0})(z-z_{0}) \end{equation*}

  Should be able to relate this to being holomorphic.

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• L’Hôpital’s rule works!

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