answer:The function f(z) = sum_{n=0}^infty z^n is the geometric series, which converges on the open unit disk D = {z in mathbb{C} : |z| < 1}. The sum of the geometric series is given by the formula: f(z) = frac{1}{1-z}, for |z| < 1. Now, we want to find the analytic continuation of f(z) to the set mathbb{C} backslash {1}. The function g(z) = frac{1}{1-z} is already analytic on mathbb{C} backslash {1}, so it serves as the analytic continuation of f(z). To find the value of g(2), we simply plug in z = 2 into the expression for g(z): g(2) = frac{1}{1-2} = frac{1}{-1} = -1. So, the value of g(2) is -1.

answer:To find the values of log(-1 + 2i) and log(-3 + 4i) using the method of analytic continuation, we first need to express the complex numbers in polar form. Recall that a complex number z = x + yi can be written in polar form as z = r(cos(θ) + i sin(θ)), where r = |z| is the magnitude of z and θ is the argument of z. 1. For z = -1 + 2i: The magnitude is |z| = r = √((-1)^2 + (2)^2) = √5. The argument is θ = arctan(Im(z) / Re(z)) = arctan(2 / -1) = arctan(-2). Since the complex number is in the second quadrant, we add π to the arctan value: θ = arctan(-2) + π. Now, we can write z in polar form: z = √5(cos(arctan(-2) + π) + i sin(arctan(-2) + π)). 2. For z = -3 + 4i: The magnitude is |z| = r = √((-3)^2 + (4)^2) = 5. The argument is θ = arctan(Im(z) / Re(z)) = arctan(4 / -3) = arctan(-4/3). Since the complex number is in the second quadrant, we add π to the arctan value: θ = arctan(-4/3) + π. Now, we can write z in polar form: z = 5(cos(arctan(-4/3) + π) + i sin(arctan(-4/3) + π)). Now, we can find the values of the logarithms using the polar forms of the complex numbers. 1. log(-1 + 2i): Using the polar form, we have log(√5(cos(arctan(-2) + π) + i sin(arctan(-2) + π))). Applying the logarithm properties, we get log(√5) + log(cos(arctan(-2) + π) + i sin(arctan(-2) + π)). The logarithm of a complex number in polar form is given by log(r) + iθ, so we have: log(√5) + i(arctan(-2) + π). 2. log(-3 + 4i): Using the polar form, we have log(5(cos(arctan(-4/3) + π) + i sin(arctan(-4/3) + π))). Applying the logarithm properties, we get log(5) + log(cos(arctan(-4/3) + π) + i sin(arctan(-4/3) + π)). The logarithm of a complex number in polar form is given by log(r) + iθ, so we have: log(5) + i(arctan(-4/3) + π). So, the values of log(-1 + 2i) and log(-3 + 4i) are: log(-1 + 2i) = log(√5) + i(arctan(-2) + π) log(-3 + 4i) = log(5) + i(arctan(-4/3) + π)

answer:To find the image of the circle |z-1| = 2 under the conformal mapping f(z) = frac{1}{2}ln frac{1+z}{1-z}, we first parameterize the circle and then apply the mapping. Let z = 1 + 2e^{itheta}, where theta in [0, 2pi). This parameterization represents the circle |z-1| = 2. Now, we apply the mapping f(z): f(z) = f(1 + 2e^{itheta}) = frac{1}{2}ln frac{1+(1+2e^{itheta})}{1-(1+2e^{itheta})} = frac{1}{2}ln frac{2+2e^{itheta}}{-2e^{itheta}} To simplify this expression, we can factor out a 2 from the numerator and denominator: f(z) = frac{1}{2}ln frac{1+e^{itheta}}{-e^{itheta}} Now, let's find the magnitude and argument of the complex number inside the logarithm: Magnitude: |1+e^{itheta}| = sqrt{(costheta + 1)^2 + (sintheta)^2} = sqrt{2 + 2costheta} |-e^{itheta}| = |-1| = 1 So, the magnitude of the complex number inside the logarithm is: frac{|1+e^{itheta}|}{|-e^{itheta}|} = sqrt{2 + 2costheta} Argument: arg(1+e^{itheta}) = arctanfrac{sintheta}{1+costheta} arg(-e^{itheta}) = theta + pi So, the argument of the complex number inside the logarithm is: argfrac{1+e^{itheta}}{-e^{itheta}} = arctanfrac{sintheta}{1+costheta} - (theta + pi) Now, we can write the image of the circle under the mapping as: f(z) = frac{1}{2}ln(sqrt{2 + 2costheta}) + frac{i}{2}left(arctanfrac{sintheta}{1+costheta} - (theta + pi)right) This expression represents the image of the circle |z-1| = 2 under the conformal mapping f(z) = frac{1}{2}ln frac{1+z}{1-z}. The image is not a simple geometric shape, but this parametric representation describes the transformed curve in the complex plane.

answer:(a) Let z = x + yi, where x and y are real numbers. Then, the given circle can be represented as: |z - 2 - i| = |(x - 2) + (y - 1)i| = 2 Now, let's find the image of this circle under the function f(z): f(z) = 4z + 3i = 4(x + yi) + 3i = (4x) + (4y + 3)i Let w = u + vi be the image of z under f(z), where u and v are real numbers. Then, we have: u = 4x v = 4y + 3 Now, we can rewrite the equation of the circle in terms of u and v: |((u/4) - 2) + ((v - 3)/4 - 1)i| = 2 Simplifying this equation, we get: |((u - 8)/4) + ((v - 7)/4)i| = 2 Now, let's multiply both sides by 4 to get rid of the fractions: |(u - 8) + (v - 7)i| = 8 Thus, the image of the circle under f(z) is another circle with the equation: |w - 8 - 7i| = 8 (b) Let's find a conformal mapping that maps the region {z: |z| < 1} onto the region {w: |w - i| < 1}. Consider the Möbius transformation: g(z) = (az + b) / (cz + d) where a, b, c, and d are complex constants. We want to find the values of a, b, c, and d such that g(z) maps the unit circle |z| < 1 onto the circle |w - i| < 1. First, let's find the image of z = 1 under g(z): g(1) = (a + b) / (c + d) Since g(1) is on the circle |w - i| < 1, we have: |g(1) - i| = 1 Now, let's find the image of z = -1 under g(z): g(-1) = (-a + b) / (-c + d) Since g(-1) is also on the circle |w - i| < 1, we have: |g(-1) - i| = 1 Now, let's find the image of z = i under g(z): g(i) = (ai + b) / (ci + d) Since g(i) is on the circle |w - i| < 1, we have: |g(i) - i| = 1 Now, we have three equations with four unknowns (a, b, c, and d). To simplify the problem, let's set d = 1 (since Möbius transformations are determined up to a constant factor, we can do this without loss of generality). Then, we have: g(z) = (az + b) / (cz + 1) Now, we have three equations with three unknowns: |((a + b) / (c + 1)) - i| = 1 |((-a + b) / (-c + 1)) - i| = 1 |((ai + b) / (ci + 1)) - i| = 1 Solving this system of equations, we find that a = 1, b = i, and c = 1. Thus, the conformal mapping g(z) is: g(z) = (z + i) / (z + 1) This mapping maps the region {z: |z| < 1} onto the region {w: |w - i| < 1}.