Aeroacoustics of Low Mach Number Flows

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=(α²−k²)^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 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

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