Complex Variables and Applications

Laplace transform solution of conjugate heat and mass transfer in falling film absorption process

“…to s iR   where is a real positive number and is larger than the real parts of all the singularities of () Fs (Fig. B.1) Letting R tend to  it can be shown that (Scott et al, 1985) lim (s) 0…”

“…The Residue theorem can be utilized to determine the inverse Laplace transform when the transform is complicated and does not appear in tables (Scott et al, 1985). As shown in Appendix B, the inverse Laplace of a complex function () Fs can be obtained by calculating sum of the residues of ( ) e st Fs at singularities of…”

“…Depending on the singularity type (simple poles, simple complex roots and repeated roots) different methods are utilized to calculate the residues (Scott et al, 1985). A (the choice and range of Le and St will be discussed later) are calculated numerically and are shown in Fig.…”

Effect of time delay on the impedance control of a pressure-based, current-driven Electroacoustic Absorber

“…It is interesting to remark that this acoustical passivity condition is totally equivalent to the passivity definition in control theory [13], where a Single-Input-Single-Output (SISO) system is defined as passive if the real part of its transfer function is positive. In our acoustic parallel, a positive real part of the mobility transfer function Y a (jω) corresponds to a positive value of the normal absorption coefficient, or also to an absolute value of the normal reflection coefficient (which is nothing less than a bilinear transform [36] of the acoustical mobility) lower than one. The minimum value of the absorption coefficient can be adopted as a passivity index, for an acoustic controlled impedance, analogously to the "Input-Feedforward-Passivity" (IFP) index defined in [29].…”

Complex Field Formulation of the Quantum Estimation Theory

“…The concept of derivation of a function of complex variables given by complex calculus [43,44] is limited to holomorphic functions, that is, functions that are infinitely differentiable in a neighborhood of the point. This is a strong restriction that leaves many interesting complex functions without derivative, for instance f (z) = |z| 2 , which plays a key role in quantum mechanics.…”