Calculus Small Changes, Big Ideas

Section 16.4 Multivalued function behavior

• Logarithmic function:

  \begin{equation*} \ln z=\ln|z|+\ii\arg z \end{equation*}

  Because the exponential function is periodic, its inverse must be multivalued. We can restrict it to arrive at the principal value

  \begin{equation*} \Ln z=\ln|z|+\ii\Arg z \end{equation*}

  Here, we use the principal argument with \(-\pii<\Arg z\le\pii\text{.}\)

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• Complex powers:

  Aside

  \begin{equation*} z^{\alpha}=\ee^{\alpha\ln z} \end{equation*}

  The principal value is \(z^{\alpha}=\ee^{\alpha\Ln z}\text{.}\)

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  For roots, we write the principal value of \(z^{1/n}\) as \(\sqrt[n]{z}\text{.}\)

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• Inverse trigonometric functions etc. can also be arrived at in terms of exponentials. For example:

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  \begin{align*} \sin\inv z \amp = -\ii\ln\qty(\ii z+(1-z^2)^{1/2})\\ \cos\inv z \amp = -\ii\ln\qty(z+\ii (1-z^2)^{1/2})\\ \tan\inv z \amp = \frac{\ii}{2}\ln\qty(\frac{\ii+z}{\ii-z})\\ \sinh\inv z \amp = \ln\qty(z+(z^2+1)^{1/2})\\ \cosh\inv z \amp = \ln\qty(z+(z^2-1)^{1/2})\\ \tanh\inv z \amp = \frac{1}{2}\ln\qty(\frac{1+z}{1-z}) \end{align*}
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• Brief look at Riemann surfaces — just enough to show where the “rest” of the function is for the functions with branch cuts.

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