Abstract
In polar coordinates \(z=re^{i\theta }\) we have: \(f(z)=f(r, \theta ) = u(r, \theta ) + i v(r, \theta )\). If we take the limit \(\Delta z = [(r+\Delta r) e^{i (\theta + \Delta \theta )} - r e^{i \theta }] \rightarrow 0\) ‘radially’, i.e. \(\Delta \theta = 0\) first and then \(\Delta r \rightarrow 0\), by definition of the derivative we obtain
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Notes
- 1.
In fact, this is the Legendre equation, see (4.6) on page 189.
- 2.
For the power function \(t^{\nu }\), the Laplace transform is given by
$$\begin{aligned} \int \limits _0^{\infty } e^{-st} t^{\nu } dt = \frac{\Gamma (\nu +1)}{s^{\nu +1}}, \qquad \; \nu > -1 \, , \end{aligned}$$which follows from definition (2.1) of the Gamma function.
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Gogolin, A.O., Tsitsishvili, E.G., Komnik, A. (2014). Solutions to the Problems. In: Tsitsishvili, E., Komnik, A. (eds) Lectures on Complex Integration. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00212-5_5
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DOI: https://doi.org/10.1007/978-3-319-00212-5_5
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