a) We solve the equation for variable (via quadratic formula), to get
Let and ; since , we have , so that are complex (and not real). Because , we find
Recall that has a power series representation:
Therefore, the above shows
Thus, letting
we obtain the power series representation
b) Since
we get
c) In part a) above we have found the power series by looking at power series of
These power series converge if and only if , respectively, . Therefore, the power series for converges if and only if . Thus, the radius of convergence is
c. Evaluate ,f(z) dz with า the circle of radius 1 centered at the origin and...
25. Show that cos eº? dz = 2ni.sin 1. 2 Z the circle of radius 8 centered a
Q5) Evaluate $c f(z) dz where C is the unit circle Iz| = 1 and f(2) is defined as follows a) f(z) = z2+z2+z_ b) f(x) = tan z c) f() = cosha
1 3. Let f(x) = 22(2-2)(2 - 4) and C a circle of radius 2k - 1 about the origin with counterclockwise orientation. (1) Find (2) Find 50, 5(=dz. Je_1(a) dz. 5. 1(a) dz. (3) Find
44. The vector field F has F 7 everywhere and C is the circle of radius 1 centered at the origin. What is the largest possible value of F dr? The smallest possible value? What conditions lead to these values? 44. The vector field F has F 7 everywhere and C is the circle of radius 1 centered at the origin. What is the largest possible value of F dr? The smallest possible value? What conditions lead to these values?
Expand the function f(z) = (z−1)/(3−z) in a Taylor series centered at the point z_0 = 1. Give the radius of convergence r of the series.
7. Let D = {z C z 1) denote the closed unit disc centered at the origin. Let f : D → C be a continuous function which is holomorphic on the interior of D. Suppose If(:) 2/(2- 2) and that If (z)1-2 for all z such that 1. Show that f(z is constant. 7. Let D = {z C z 1) denote the closed unit disc centered at the origin. Let f : D → C be a continuous...
1+ z Expand the function f(z) = in a Taylor Series Centered at Zo=i. Write the full series i.e., all the terms. Use The Sigma Notation. Find the radius R of the Disk of Convergence centered at zo.
Evaluate the integral $ 3z + 4)cos z dz, C:12+2i = 1 counterclockwise. z² +4 Integrate $c +436 yosz dz, C:/z+3–2i = 3 counterclockwise. (z +4)
4. Evaluate the following integrals: f, where contour γ is a circle of radius 2 centered at the origin. z.İ f, -1-i,1-i,1+i,and-1+i. (z-0.1-1); where contour γ is the square with the four vertices ill) Jo (2+7 cos(e)) 4. Evaluate the following integrals: f, where contour γ is a circle of radius 2 centered at the origin. z.İ f, -1-i,1-i,1+i,and-1+i. (z-0.1-1); where contour γ is the square with the four vertices ill) Jo (2+7 cos(e))
4. Evaluate the following integrals: f, where contour γ is a circle of radius 2 centered at the origin. z.İ f, -1-i,1-i,1+i,and-1+i. (z-0.1-1); where contour γ is the square with the four vertices ill) Jo (2+7 cos(e)) 4. Evaluate the following integrals: f, where contour γ is a circle of radius 2 centered at the origin. z.İ f, -1-i,1-i,1+i,and-1+i. (z-0.1-1); where contour γ is the square with the four vertices ill) Jo (2+7 cos(e))