Cube roots

Cube roots

Complex numbers

Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017

Thus |z|3 = 27 and 3iθ = 2nπi. When n = 0 we have θ = 0; when n = 1 we have θ=2π3; when n = 2we have θ=4π3. It is instructive to write the complex cube roots of 1 as –1±i32. The three cube roots of 27 are z = 3 and z=32(–1+i3) and z=32(–1–i3).

Formulas for the H/V ratio (ellipticity) of Rayleigh waves in orthotropic elastic half-spaces

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Journal Information

Published in Waves in Random and Complex Media, 2019

Pham Chi Vinh, Tran Thanh Tuan, Le Thi Hue

Proof Suppose inequalities (8) hold. According to Chadwick [22] (see also Barneet and Lothe [18,19]), there exists a unique Rayleigh wave propagating along the -direction and attenuating in the -direction. Now we prove that its H/V ratio is given by formulas (31), (33) and (34).Let , then in terms of z Equation (25) takes the form: where: According to the theory of cubic equation, three roots of Equation (35) are calculated by: where: In relation to the formulas (38), we emphasize two points:The cube root of a negative real number is taken as the real negative root.If, in the expression S, is complex, the phase angle in T is taken as the negative of the phase angle in S so that where is the complex conjugate of S.

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Complex numbers

Formulas for the H/V ratio (ellipticity) of Rayleigh waves in orthotropic elastic half-spaces