By Daniel Tanre
ISBN-10: 3540127267
ISBN-13: 9783540127260
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Additional resources for Homotopie Rationnelle. Modeles de Chen Quillen Sullivan
Example text
G = ax. au =u(z) + iv(z) satisfies the Cauchy-Riemann equaD We have thus given the desired characterization of harmonic functions as real parts of holomorphic functions, at least when the functions are defined on an open rectangle or open disc. 2 do indeed hold when R is replaced by an open disc (or by any convex open set). 1 is essentially an antidifferentiation theorem. Given our new notation, it would be aesthetically pleasing if we could use it to solve the following antidifferentiation problem: If F is holomorphic, can we find a holomorphic function H such that oHI {jz = F?
Cauchy's Theorem and Formula 43 except possibly at P1 (the first coordinate of P). 1, d dx H(x) = F(x) for all x; hence {}h ox =f as desired. 3. 3. Let U ~ C be either an open rectangle or an open disc. Let P E U be fixed. Suppose that F is continuous on U and holomorphic on U \ {P}. Then there is a holomorphic H on U such that 8Hjoz =F. Proof. 1. 3 is the main and indeed the only point of this section. We shall apply it in the next section as follows: f will be a holomorphic function on U, and z E U will be fixed.
Complex Line Integrals Such a choice of () is possible because (y'a2a+b2)2 + (y'a2b+b2)2 = 1. Thus the Cauchy-Riemann equations imply that the (real) Jacobian of has the form ( ~ ~ ) . ( ~~:: - f ~~:: ) for some ,\ E IR, ,\ > 0, and some () E R Geometrically, these two matrices have simple meanings. The matrix () ( cos sin() - sin () ) cos() is the representation of a rotation around the origin by the angle (). A. Therefore the product ( ~ ~ ) . ( ~~:: - ~~:: ) represents the same operation on IR 2 as does multiplication on C by the complex number ,\ei8 .
Homotopie Rationnelle. Modeles de Chen Quillen Sullivan by Daniel Tanre
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