1. What is a Riemann sheet (define it, and detail its properties) 2. Describe the Riemann...

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1. What is a Riemann sheet (define it, and detail its properties) 2. Describe the Riemann sheet of G(z) = ln(z). 3. Describe with a figure and then discuss the branch cut for f(z) = log(z). 4. Describe the branch cut for f(z) = e². 5. Find f(z) and f'(z) at zo = 0 and 2₁ = eiñ/² = i for f(2)=√1-2² and describe the branch-cuts required to make this function single valued. 6. If w = F(s) = 1+s² (a) What is s = G(w) = F-¹(w). (b) Map out the range of s = G(w) for the two domains of w. Explain. (c) Discuss reasonable places to place the branch cut(s)? 7. Starting with the formula for the impedance of a transmission line (TL): z(s) =tan(s) show that 1 u+jv=atan((x+jy)) tan−¹(2): -log u+jv = (1/2)*log((1-i*(x+jy))/(1+i*(x+jy))) mo X (3) Figure 2: This figure compares the colorized plots of w = tan-¹(2) and w = = ln(1 − iz)/(1 + iz), to verify they are the same function. 1. What is a Riemann sheet (define it, and detail its properties) 2. Describe the Riemann sheet of G(z) = ln(z). 3. Describe with a figure and then discuss the branch cut for f(z) = log(z). 4. Describe the branch cut for f(z) = e². 5. Find f(z) and f'(z) at zo = 0 and 2₁ = eiñ/² = i for f(2)=√1-2² and describe the branch-cuts required to make this function single valued. 6. If w = F(s) = 1+s² (a) What is s = G(w) = F-¹(w). (b) Map out the range of s = G(w) for the two domains of w. Explain. (c) Discuss reasonable places to place the branch cut(s)? 7. Starting with the formula for the impedance of a transmission line (TL): z(s) =tan(s) show that 1 u+jv=atan((x+jy)) tan−¹(2): -log u+jv = (1/2)*log((1-i*(x+jy))/(1+i*(x+jy))) mo X (3) Figure 2: This figure compares the colorized plots of w = tan-¹(2) and w = = ln(1 − iz)/(1 + iz), to verify they are the same function.