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Appendix B

# Branch cuts

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=(α2k2)1/2=(−(s2+k2))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 ko. 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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