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# HW8 - Explain(c If F s = log s where are the Cauchy-Riemann...

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ECE 493 HW #8 – Version 1.2 January 25, 2011 Spring 2011 Univ. of Illinois Due Tu, Jan 25 Prof. Allen Topic of this homework: Analytic functions of a complex variable; Deliverable: Please, show your work. 1. Complex functions: Domain: s σ + , Range: Z ( s ) R ( s ) + iX ( s ). The Domain (e.g., s ) and Range (e.g., Z ( s )) are described in the text on page 1114. Plot the Range Z ( s ) in terms of the specified Domain in s . (a) Domain: s = σ , Range: Z ( s ) = 1 + s . (b) Domain: s = , Range: Z ( s ) = 1 + s . (c) Domain: s = , Range: H ( s ) = 1 + s 2 . (d) Reverse the range and domain. Thus the Domain is H ( s ) = 1 + s 2 while the Range s . Plot the range for H = R . 2. Find the solutions (numerical values in the form a + ib ) of the following: (a) x 2 + 1 = 0 (b) x 3 + 8 = 0 (c) i i (Show your work, as always!) (d) What is the frequency of a t for any constant a ? 3. Harmonic functions (a) Show that if F ( s ) = e s that the real and imaginary parts obey the Cauchy-Riemann conditions. (b) If F ( s ) = s/ (1 + s ), where are the Cauchy-Riemann conditions valid, or not valid? Explain.
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Unformatted text preview: Explain. (c) If F ( s ) = log( s ), where are the Cauchy-Riemann conditions valid, or not? Explain. (d) If F ( s ) = √ 1 + s 2 , where are the Cauchy-Riemann conditions valid, or not? Explain. 4. Branch cuts (a) If w ≡ F ( s ) = 1 + s 2 i. What is s = G ( w ) ≡ F-1 ( w ). ii. Map out the range of s = G ( w ) for the two domains of w . Explain. iii. Discuss reasonable places to place the branch cut(s)? (b) Describe the Riemann surface of G ( z ) = ln( z ). 5. Laplace transforms (a) ±ind the Laplace transform of 1, d/dt , i t-∞ δ ( t ) dt , and i t-∞ u ( t ) dt . . (b) If f ( t ) = 1 / √ πt has a Laplace transform F ( s ) = 1 / √ s : i. What is the inverse Laplace transform of √ s ? ii. what is f (-1)? Version 1.2 January 25, 2011 ˜ /493/Assignments/HW #8 – Version 1.2 January 25, 2011...
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