- 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**∈ Cand its complex conjugate^{n }**z**^{∗ }=**x**−j·**y**∈ Cwith^{n }**x***,***y**∈ R. We have that the total differential of^{n}*g*and*f*, respectively, is

T

d*g *=d**z**^{∗ }(2) TT

d*f *=d**x**d**y***. *(3)

- By using the fact that d
*g*= d*f*, show that

(4)

*. *(5)

2

- From the previous result show that

(6)

*. *(7)

- If
*f*(*x,y*) =*u*(*x,y*)+j*v*(*x,y*), where*u*(*x,y*)*,v*(*x,y*) ∈ R show that the differential d*g*does not depend on the differential d**z**^{∗ }if*g*(**z***,***z**^{∗}) =*f*(**x***,***y**) is analytic, i.e. show that.

- Consider the function

*I*(**w***,***w**^{∗}) = **w**^{H}**Rw*** ,*

with **w***,***p **∈ C* ^{n }*and

**R**=

**R**

*∈ C*

^{H }

^{n}^{×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 **∈ C^{3}*,f *∈ R*,***g **∈ C^{2 }by means of complex Lagrangian multipliers.

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