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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
A complex function f is analytic on an open set U if it has a derivative at each point of U. The function f is said to be analytic at the point z0 if it is analytic on some open set containing z0. A function analytic on all of the complex plane is an entire function. If f(z)=u(x,y)+iv(x,y) is analytic in an open set U, then the real and imaginary parts of f(z) satisfy the Cauchy-Riemann equations: ∂u∂x=∂v∂y and ∂u∂y=-∂v∂x
Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
This series converges uniformly and absolutely inside the circle of convergence, c|z − z0| < R, with radius of convergence R. [See the Appendix for a review of convergence.] There are various types of complex-valued functions.A holomorphic function is (complex) differentiable in a neighborhood of every point in its domain.An analytic function has a convergent Taylor series expansion in a neighborhood of each point in its domain. We see here that analytic functions are holomorphic and vice versa.If a function is holomorphic throughout the complex plane, then it is called an entire function.Finally, a function which is holomorphic on all of its domain except at a set of isolated poles (to be defined later), then it is called a meromorphic function.
Some mathematical tools
Published in Bertero Mario, Boccacci Patrizia, Introduction to Inverse Problems in Imaging, 2020
Bertero Mario, Boccacci Patrizia
function is an entire function, i.e a function which is analytic everywhere. But an analytic function, which vanishes on some interval, does also vanish everywhere. As a consequence, if /(x) is spacelimited and not identically zero, then the support of /(") must be the whole axis (-00, +00). The second part of the statement follows from the symmetry between the FT and the inverse FT. Moreover the statement can be generalized to functions of two and more variables.
Remarks on the inverse problem for an energy-dependent hamiltonian
Published in Applicable Analysis, 2023
Pierre Chau Huu-Tai, Bernard Ducomet
As ϕ is an entire function, using residue theorem in the second integral in the left-hand side, we get As is holomorphic for we see that and using Equation (13) we find Using (45) we get finally Now we consider for Pushing to the real axis (using Proposition 2.2) one checks that the integral in the right-hand side of (53) gives by using inverse Fourier transform.
Moving semi-infinite crack between dissimilarorthotropic strips
Published in Waves in Random and Complex Media, 2022
Subhadeep Naskar, S. C. Mandal
Using (54), Equation (53) after rearrangement becomes It can be seen that the left-hand side of Equation (56) is analytic for and right-hand side is analytic for for any arbitrary small positive ε. Using the theorem of analytic continuation, we can conclude that is the common region of analyticity. Hence, Equation (56) is analytic and single valued throughout the complex plane, which signifies the modification of the constant load in Equation (31). Therefore, both sides of Equation (56) are equal to a common entire function since through the whole complex plane both sides of Equation (56) are analytic.
Inverse problems for the impulsive Sturm–Liouville operator with jump conditions
Published in Inverse Problems in Science and Engineering, 2019
Yasser Khalili, Nematollah Kadkhoda, Dumitru Baleanu
Define where This function is an entire function in ρ. For convenience, we denote By the assumption of the theorem, we will have Since the characteristic function is an entire function of order we infer for some positive constant C. By standard calculations (see [21,24]), it is concluded that Since is a complete set of eigenvalues for L, we can write for sufficiently large b Therefore, by (42) and (43), we have Taking (36), (37) and (44), we get By Phragmén–Lindelöf theorem, we have for all ρ and then for all ρ.