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![(O ſto) dh rfi) 12+ integration with contour e -- 0 2+1 R Consider the Complen c= [-R R] U TR (2) dz. с. fcz) f(2) F (Z) = tr](http://img.homeworklib.com/questions/bc32c2e0-899d-11eb-b7da-3328feec5836.png?x-oss-process=image/resize,w_560)
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Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed...
Problem 4. (5 points) Suppose f is analytic on and inside a simple closed curve C....
Problem 4. (5 points) Suppose f is analytic on and inside a simple closed curve C. Assume f(x) = 0 for z on C. Show f(2)=0 for all z inside C.
9. Suppose that f (z) has a simple pole at ao on a closed curve C,...
9. Suppose that f (z) has a simple pole at ao on a closed curve C, but is analytic elsewhere inside and on C except for poles at a finite number of interior points a1,a2,, (a) If the contour C is indented at ao by a circular arc with center at ao, show that the limiting form of the integral of f (x) around the indented contour is as the radius of the indentation tends to zero, regardless of whether...
Question 5. Let f(2) = for z e H4 = {z : Im z > 0},...
Question 5. Let f(2) = for z e H4 = {z : Im z > 0}, the open upper half-plane of C. 2+i [2]a) Show that f maps H4 into the open unit disc |2| < 1. Hint: compute |f(2)|² for z e H4. [3]b) Show that ƒ maps the boundary of H onto the boundary of the disc |2| <1 minus one point. What point is missed?
Q5. a) Let f(z) be an analytic function on a connected open set D. If there...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...
Let G be an open subset of C. Suppose f: GC is analytic except for isolated...
Let G be an open subset of C. Suppose f: GC is analytic except for isolated poles. Show that the poles of f can not have a limit point in G.
Let C be the closed curve defined by r(t) = costi + sin tj + sin...
Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) (5 pts] Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts] By using Stokes' Theorem, evaluate the line integral F. dr C where F(x, y, z) = (y2 + cos x)i + (sin y +22)j + xk
4. Let C be the closed curve defined by r(t) = costi + sin tj +...
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t<2n. (a) [5 pts] Show that this curve C lies on the surface S defined by z = 2.cy. (b) [20 pts] By using Stokes’ Theorem, evaluate the line integral| vi F. dr where F(x, y, z) = (y2 + cos x)i + (sin y + z2)j + xk
4. Let C be the closed curve defined by r(t) = costi + sin tj +...
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) [5 pts) Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts) By using Stokes' Theorem, evaluate the line integral F. dr с where F(x, y, z) = (y2 + cos x)i + (sin y + z2)j + xk
Miscellaneous Exercises 253 d an analytic (integral) solution of this wave problem 6 au 1 a2u -00
Miscellaneous Exercises 253 d an analytic (integral) solution of this wave problem 6 au 1 a2u -00
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d)...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
Problem 4. (5 points) Suppose f is analytic on and inside a simple closed curve C. Assume f(x) = 0 for z on C. Show f(2)=0 for all z inside C.
9. Suppose that f (z) has a simple pole at ao on a closed curve C, but is analytic elsewhere inside and on C except for poles at a finite number of interior points a1,a2,, (a) If the contour C is indented at ao by a circular arc with center at ao, show that the limiting form of the integral of f (x) around the indented contour is as the radius of the indentation tends to zero, regardless of whether...
Question 5. Let f(2) = for z e H4 = {z : Im z > 0}, the open upper half-plane of C. 2+i [2]a) Show that f maps H4 into the open unit disc |2| < 1. Hint: compute |f(2)|² for z e H4. [3]b) Show that ƒ maps the boundary of H onto the boundary of the disc |2| <1 minus one point. What point is missed?
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...
Let G be an open subset of C. Suppose f: GC is analytic except for isolated poles. Show that the poles of f can not have a limit point in G.
Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) (5 pts] Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts] By using Stokes' Theorem, evaluate the line integral F. dr C where F(x, y, z) = (y2 + cos x)i + (sin y +22)j + xk
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t<2n. (a) [5 pts] Show that this curve C lies on the surface S defined by z = 2.cy. (b) [20 pts] By using Stokes’ Theorem, evaluate the line integral| vi F. dr where F(x, y, z) = (y2 + cos x)i + (sin y + z2)j + xk
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) [5 pts) Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts) By using Stokes' Theorem, evaluate the line integral F. dr с where F(x, y, z) = (y2 + cos x)i + (sin y + z2)j + xk
Miscellaneous Exercises 253 d an analytic (integral) solution of this wave problem 6 au 1 a2u -00
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
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