Appendix B

We have frequently referred to choosing the correct branch cut to fully define a multivalued function. For example, in Chapter 13 we have required that the real part of γ=(α^{2}−k^{2})^{1/2}=(−(s^{2}+k^{2}))^{1/2} and the imaginary part of $\kappa ={\left({k}_{o}^{2}-{\nu}^{2}\right)}^{1/2}$ be positive. These are multivalued functions and so they can only be fully defined for restricted values of the complex variables s, α, ν, or k_{o}. The purpose of this appendix is to fully explain what is implied by these conditions.

First we note that the square root of a complex number z is given by

${z}^{1/2}={r}^{1/2}{e}^{i\theta /2}$

and is only completely defined if θ is restricted to the range

$\vartheta <\theta <\vartheta +2\pi $

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