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(a) Find functions u, v: R² → R such that g(x+iy) = u(x, y) + iv(x, y) for all x, y EE (b) State the Cauchy-Riemann equations, and show that u and us tions everywhere in R2. (c) Use the Cauchy-Riemann equations to show that gʻ(z) = −g(z)

(a) Find functions u, v: R² → R such that g(x+iy) = u(x, y) + iv(x, y) for all x, y EE (b) State the Cauchy-Riemann equations, and show that u and us tions everywhere in R2. (c) Use the Cauchy-Riemann equations to show that gʻ(z) = −g(z)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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Let g: C→C be defined by g(z) = exp(-z). (a) Find functions u, v : R² → R such that g(x+iy) = u(x, y) + iv(x, y) for all x, y E R. (b) State the Cauchy-Riemann equations, and show that u and v satisfy these equa- tions everywhere in R². (c) Use the Cauchy-Riemann equations to show that g′(z) = −g(z).
Transcribed Image Text:Let g: C→C be defined by g(z) = exp(-z). (a) Find functions u, v : R² → R such that g(x+iy) = u(x, y) + iv(x, y) for all x, y E R. (b) State the Cauchy-Riemann equations, and show that u and v satisfy these equa- tions everywhere in R². (c) Use the Cauchy-Riemann equations to show that g′(z) = −g(z).
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