Complex 6. Find a branch of the multiple-valued expression f(z)-log(i(z + 21)) which is analytic for...
6. Determine the sets on which the following functions are analytic. a. f(z) = Log(z + 1) b. f(x) = Log(x+)
5. 15 points Find a branch of log(z2 + iz – 3) such that it is analytic at z = i, and find its derivative at z =i.
Problem 1 Consider the composition f(w(z)) of two complex valued functions of a complex variable, f(w) and w(z), where z = x+iy and w=u+iv. Assume that both functions have continuous partial derivatives. Show that the chain rule can be written in complex form as of _ of ou , of Oz . . of az " dw dz * dw dz and Z of ou , of ou dw dz* dw ƏZ Show as a consequence that if f(w) is...
Let p(z) be the principal branch of 21-i. Let D* = C\(-00,0) be all the complex numbers except for the non-positive real numbers. (a) (4 points) Find a function which is an antiderivative of p(x) on D". (b) (6 points) Let I be a contour such that (i) I is contained in D* and (ii) the initial point of I' is 1 and the terminal point of I is i. Compute (2)dr. Justify your answers.
2. (20 points) (a). Find the derivative of the complex-valued function f() = (2-21) (3z + 5) at z = 31.
14. Determine whether or not the complex function f(z) -4x2+5x -4y2 + i(8xy 5y+3) is analytic for all z є c.
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
56. Consider the multiple-valued fu nction F()z1/3 that assigns to z the set of three cube roots of z. Explicitly define three distinct branches fi. fa, and fs of F, all of which have the nonnegative real axis as a branch cut.
6. (a) Let R > 0 and zo E C. If f is analytic in the disk (z – zol < R and f(k)(zo) = 0 for all k= 1,2,3,..., show that f is constant and is completely determined by its value at zo.
1. if the real part of an analytic function, f(z), is given find the imaginary part, v(x, y) and f(z) as a function of x. (step by step) 2. Evaluate the following complex integral (step by step) 1. If the real part of an analytic function, f(z), is given as 2 - 12 (x2 + y2)2 find the imaginary part, v(x,y), and f(z) as a function of z. 2. Evaluate the following complex integral: