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Aeroacoustics of Low Mach Number Flows by William Devenport, Stewart Glegg

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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 κ=ko2ν21/2si2_e 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

z1/2=r1/2eiθ/2

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

ϑ<θ<ϑ+2π

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