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## David R. Steward

Print publication date: 2020

Print ISBN-13: 9780198856788

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780198856788.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 28 November 2021

# Analytic Elements from Complex Functions

Chapter:
(p.103) 3 Analytic Elements from Complex Functions
Source:
Analytic Element Method
Publisher:
Oxford University Press
DOI:10.1093/oso/9780198856788.003.0003

The mathematical functions associated with analytic elements may be formulated using a complex function $\Omega$ of a complex variable ${\zcomplex}$. Complex formulation of analytic elements is introduced in Section 3.1 for exact solutions obtained by embedding point elements that generate divergence, circulation, or velocity within a uniform vector field. Influence functions for analytic elements with circular geometry are obtained using Taylor and Laurent series expansions in Section 3.2, and conformal mapping extends this formulation to analytic elements with the geometry of ellipses (Section 3.3). The Courant's Sewing Theorem is employed in Section 3.4 to develop solutions for interface conditions across straight line segments, and the Joukowsky transformation extends methods to circular arcs and wings (Section 3.5), which satisfy a Kutta condition of non-singular vector field at their trailing edges. Vector fields with spatially distributed divergence and curl are formulated using the complex variable ${\zcomplex}$ with its complex conjugate $\overline{\zcomplex}$ in Section 3.6, and the complex conjugate is further employed in the Kolosov formulas (Section 3.7) to solve force deformation problems for analytic elements with traction or displacement specified boundary conditions.

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