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- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/09%3A_Residue_TheoremThumbnail: Illustration of the setting. (Public Domain; Ben pcc via Wikipedia)
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/10%3A_Definite_Integrals_Using_the_Residue_Theorem/10.08%3A_Solving_DEs_using_the_Fourier_transform\(\begin{array} {ccl} {\lim_{R \to \infty} \int_{C_R} e^{izt} g(z)\ dz = 0} & \ \ \ \ \ \ & {\text{(Theorem 10.2.2(b))}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_2} e^{izt} g(z) \ dz = \pi i \text{Res...\(\begin{array} {ccl} {\lim_{R \to \infty} \int_{C_R} e^{izt} g(z)\ dz = 0} & \ \ \ \ \ \ & {\text{(Theorem 10.2.2(b))}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_2} e^{izt} g(z) \ dz = \pi i \text{Res} (e^{izt} g(z), -1)} & \ \ \ \ \ \ & {\text{(Theorem 10.7.2)}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_4} e^{izt} g(z)\ dz = \pi i \text{Res} (e^{izt} g(z), 1)} & \ \ \ \ \ \ & {\text{(Theorem 10.7.2)}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_1 + C_3 + C_5} e^{izt} g(z) \ dz = \text{p.v.} \hat{y…
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/02%3A_Analytic_Functions/2.09%3A_Branch_Cuts_and_Function_CompositionWe often compose functions, i.e. f(g(z)) . In general in this case we have the chain rule to compute the derivative. However we need to specify the domain for z where the function is analytic. And ...We often compose functions, i.e. f(g(z)) . In general in this case we have the chain rule to compute the derivative. However we need to specify the domain for z where the function is analytic. And when branches and branch cuts are involved we need to take care.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/10%3A_Definite_Integrals_Using_the_Residue_Theorem/10.01%3A_Integrals_of_functions_that_decayThe theorems in this section will guide us in choosing the closed contour C described in the introduction.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/13%3A_Laplace_Transform/13.03%3A_Exponential_TypeThe Laplace transform is defined when the integral for it converges. Functions of exponential type are a class of functions for which the integral converges for all s with Re(s) large enough.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/07%3A_Two_Dimensional_Hydrodynamics_and_Complex_Potentials/7.06%3A_More_Examples_with_Pretty_PicturesΦ(z)=log(z−1)+log(z+1)=log((z−1)(z+1))=log(z2−1). Now, as z approaches the y-axis from one side or the other, the argument of log(z2−1) a...Φ(z)=log(z−1)+log(z+1)=log((z−1)(z+1))=log(z2−1). Now, as z approaches the y-axis from one side or the other, the argument of log(z2−1) approaches either π or −π. Farther away from the origin the flow stops being radial and is pushed to the right by the uniform flow. It is the point on the x-axis where the flow from the source exactly balances that from the uniform flow.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/08%3A_Taylor_and_Laurent_Series/8.01%3A_Geometric_SeriesHaving a detailed understanding of geometric series will enable us to use Cauchy’s integral formula to understand power series representations of analytic functions. We start with the definition:
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/13%3A_Laplace_TransformThe Laplace transform takes a function of time and transforms it to a function of a complex variable s . Because the transform is invertible, no information is lost and it is reasonable to think of a...The Laplace transform takes a function of time and transforms it to a function of a complex variable s . Because the transform is invertible, no information is lost and it is reasonable to think of a function f(t) and its Laplace transform F(s) as two views of the same phenomenon. Each view has its uses and some features of the phenomenon are easier to understand in one view or the other.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/11%3A_Conformal_Transformations/11.05%3A_Riemann_Mapping_TheoremThe Riemann mapping theorem is a major theorem on conformal maps.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/00%3A_Front_Matter/02%3A_InfoPageThe LibreTexts libraries are Powered by NICE CXOne and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the Californi...The LibreTexts libraries are Powered by NICE CXOne and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/03%3A_Multivariable_Calculus_(Review)/3.07%3A_Green's_TheoremC must be piecewise smooth (traversed so interior region R is on the left) and piecewise smooth (a few corners are okay). Figure \PageIndex1: Examples of piecewise smooth and piecewise s...C must be piecewise smooth (traversed so interior region R is on the left) and piecewise smooth (a few corners are okay). Figure \PageIndex1: Examples of piecewise smooth and piecewise smooth regions. (CC BY-NC; Ümit Kaya) Theorem \PageIndex1: Green's Theorem If the vector field F=(M,N) is defined and differentiable on R then ∮CF⋅dr=∫∫RcurlF dA. where the curl is defined as curlF=(Nx−My)
