MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION A) 1. (10 points) Calculate 3+i √ √ 2 − 2i + |1/ 2 + i/ 2| Solution: 7 9 + i 13 13 2. (10 points) Prove that Re z ≤ |z| for any complex number z. Solution: Re z = x ≤ |x| ≤ p x2 + y 2 = |z| 3. (10 points) Find the roots z of z4 + 9 = 0 Solution: ± 1√ 1√ 6± 6i 2 2 4. (10 points) What is the value of the limit z2 ? z→0 |z|2 lim Explain your answer. Solution: If z = x is real, then you get x2 /x2 = 1 while if z = iy then you get −y 2 /y 2 = −1 so the limit does not exist as z → 0. Date: February 6, 2001. 1 2 MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION A) 5. (10 points) Find all complex numbers z which solve the equation √ 2 = cos z . Solution: Using the definition cos z = eiz + e−iz 2 we find the equation √ eiz + e−iz = 2 2 . Try writing q = eiz . Then our equation is √ 1 =2 2 q so if we multiply by q then √ q 2 + 1 = 2 2q q+ a quadratic equation. Solve it to find √ q = 2±1. But now return to q = eiz to find z: z = −i Log q or √ z = −i ln( 2 ± 1) + 2πN for any integer N . 6. (10 points) What are all of the values of ii ? Solution: By definition ii = exp(i Log i) and so we work out Log i = i π + 2πN 2 where N can be any integer. Then we find π ii = exp − − 2πN . 2 MATH 3160: APPLIED COMPLEX VARIABLES TEST #1 (VERSION A) 3 7. (10 points) Bonus: Derive the polar coordinate expressions of the Cauchy–Riemann equations from the rectangular ones given on page 1 above. Solution: We will use the equations x = r cos θ y = r sin θ as follows: u r uθ ux = vr vθ vx ux = vx xr xθ yr yθ uy cos θ −r sin θ vy sin θ r cos θ uy vy Therefore to solve for the rectangular coordinate derivatives: −1 u x uy ur uθ cos θ −r sin θ = vx vy vr vθ sin θ r cos θ cos θ sin θ ur uθ = vr vθ − sinr θ cosr θ ur cos θ − 1r uθ sin θ ur sin θ + 1r uθ cos θ = vr cos θ − 1r vθ sin θ vr sin θ + 1r vθ cos θ Then the usual Cauchy–Riemann equations say 1 1 ur cos θ − uθ sin θ = vr sin θ + vθ cos θ r r 1 1 ur sin θ + uθ cos θ = − vr cos θ − vθ sin θ r r Another way to write this: 1 ur = v θ r 1 v r = − uθ r