Description
- A complex valued function f(z) ∈ C of a complex valued argument z ∈ C can always be expressed in terms of two real valued functions u(x,y),v(x,y) ∈ R of two real-valued variables x,y ∈ R:
f(z) = f(x + j · y) = u(x,y) + j · v(x,y).
In the following u(x,y),v(x,y) are to be continuously differentiable with respect to x and y in an arbitrarily small region around z. The complex derivative of f(z) with respect to z is defined as
(1)
- Write (1) in terms of ∂u/∂x and ∂v/∂x by using ∆z = ∆x, i.e. by moving parallel to the real axis to the point z.
- Repeat the exercise using ∆z = j ∆y, i.e. by moving parallel to the imaginary axis to the point z.
- In order for (1) to be uniquely defined, these two results must be the same. Whatconstraint does this impose on u(x,y) and v(x,y) ?
- Compare this result to the Cauchy-Riemann equations.
- Let g(z,z∗) = f(x,y) ∈ C be a function of a complex vector z = x + j y ∈ Cn and its complex conjugate z∗ = x−j·y ∈ Cn with x,y ∈ Rn. We have that the total differential of g and f, respectively, is
T
dg =dz∗ (2) TT
df =dxdy. (3)
- By using the fact that dg = df, show that
(4)
. (5)
2
- From the previous result show that
(6)
. (7)
- If f(x,y) = u(x,y)+jv(x,y), where u(x,y),v(x,y) ∈ R show that the differential dg does not depend on the differential dz∗ if g(z,z∗) = f(x,y) is analytic, i.e. show that.
- Consider the function
I(w,w∗) = wHRw ,
with w,p ∈ Cn and R = RH ∈ Cn×n.
- Is I(w,w∗) a real valued function?
- Find a w that minimizes I(w,w∗) by solving.
- Find a w that minimizes I(w,w∗) by solving.
- Compare the results of 3b and 3c.
- Solve the following constrained real-valued minimization problem
minimize (8)
| subject to g(x1,x2) = 1 + x1 − 2x2 = 0 x1,x2,f,g ∈ R,
(a) by solving (9) for x2 in terms of x1 and then minimizing (8). (b) by means of (real) Lagrangian multipliers. 5. Solve the following constrained complex minimization problem: |
(9) |
minimize w (10)
1 −j H
subject to g(w) = j 2 w, (11) 1 j
with w ∈ C3,f ∈ R,g ∈ C2 by means of complex Lagrangian multipliers.
