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Electromagnetic Waves
Published in Myeongkyu Lee, Optics for Materials Scientists, 2019
The complex number Aeiθ is represented in the complex plane by a vector of magnitude A inclined at an angle θ to the real axis. As θ increases, this vector is counterclockwise rotated and the real part of the complex number changes between A and −A. The wave given by eq 1.24 has a disturbance varying between these two values. Accordingly, it is quite common to write the harmonic wave function as () ψ(x,t)=Aei(kx−ωt)=Aeiθ,
Signals and their functional representation
Published in Alexander D. Poularikas, ®, 2018
If z1 = 2 + j and z2 = 1 − 3j, then z = z1 + z2 = 3 − 2j. The Cartesian representation of these complex numbers and their sum is shown in Figure A1.2.1. The plane on which the complex numbers are presented is known as the complex plane. Observe that complex numbers add in the complex plane as forces add. This is also true for subtraction of complex numbers. Figure A1.2.2 presents the following forms of a complex number: z = 2 + j; z* = conjugate of z = 2 − j; − z = reflected of z; −z* reflected of its conjugate form; z = (22 + 12)1/2 exp(jtan−1(1/2)) = polar form.
Introduction to Vibrations and Waves
Published in J. David, N. Cheeke, Fundamentals and Applications of Ultrasonic Waves, 2017
where j=−1. Generally, j is used in engineering practice and i in mathematics and physics, but this is not universal. When they are not used as an index, the scalars i or j always represent −1. We may use them interchangeably. In the complex plane, the x axis represents the “real” part and the y axis represents the “imaginary” part of a variable z = x + iy = reiθ. When a physical quantity is represented by a complex variable z, by convention its physically significant part is given by Re(z). This is pure convention; since the real and imaginary parts contain redundant information, the imaginary part could equally well have been chosen. The semantics have been chosen to reinforce the conventional choice.
A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problems
Published in Journal of the Chinese Institute of Engineers, 2020
Jeng-Tzong Chen, Ying-Te Lee, Jia-Wei Lee, Sheng-Kuang Chen
To avoid calculating the Riemann principal value (RPV) and Cauchy principal value (CPV) in the BIEM, a powerful mathematical tool, degenerate kernel, is introduced to analytically study. To derive the degenerate kernel of fundamental solution in terms of the polar coordinates, a source point s and a field point x can be represented as and , respectively. In the theory of complex variables, a point z in the Argand plane (complex plane) can be represented as and . Then, the separable logarithmic function in the fundamental solution can be found for the 2D Laplace problem in Chen et al. (2014) as follows: