Use Cauchy-Riemann equations to show that the complex function
f(z) = f(x + iy) = z(x + iy) is nowhere differentiable except at
the origin z = 0.6 points) 2. Use Cauchy's theorem to evaluate the
complex integral ekz -dz, k E R. Use this result to prove the
identity 0"ck cos θ sin(k sin θ)de = 0
Using Newton-Raphson method, find the complex root of the
function f(z) = z 2 + z + 1 with with an accuracy of 10–6. Let z0 =
1 − i. write program c++ or matlab
show that a 2x2 complex matrix A is nilpotent if and only if
Tr(A)=0 and Tr(A^2)=0. give an example of a complex 2x2 matrix
which is not nilpotent but whose trace is 0
Show that the set of all solutions to the differential
equation
f′′+f=0,
where f∈F(R), is a real vector space with respect to the usual
definition of vector addition and scalar multiplication of
functions
Show that W = {f : R → R : f(1) = 0} together with usual
addition and scalar multiplication forms a vector space. Let g : R
→ R and define T : W → W by T(f) = gf. Show that T is a linear
transformation.
Let f be a differentiable function on the interval [0, 2π] with
derivative f' . Show that there exists a point c ∈ (0, 2π) such
that cos(c)f(c) + sin(c)f'(c) = 2 sin(c).
Suppose a function f : R → R is continuous with f(0) = 1. Show
that if there is a positive number x0 for which
f(x0) = 0, then there is a smallest positive number p
for which f(p) = 0. (Hint: Consider the set {x | x > 0, f(x) =
0}.)