Matematicas Avanzadas

1.- f(z)=z-i

= x+iy - i

=x+ i(y-1)

U(x,y)= x

V(x,y)= y-1

du/dx=1 du/dy=0

dv/dx=0 dv/dy=1

∴cuple con las ecuaciones de Cauchy-Riemann

2.- f(z)=z^2-iz

=〖(x+iy)〗^2-i(x+iy)

=x^2+2xiy-y^2-xi-i^2 y

=x^2-y^2+y+i(2xy-x)

u(x,y)=x^2-y^2+y

v(x,y)=2xy-x

∂u/∂x=2x ∂u/∂y=-2y+1

∂v/∂x=2y+1 ∂v/∂y=2x

∴si cumple con las ecuaciones de Cauchy-Riemann

3.- f(z)= |z|

=√(x^2+y^2 )

U(x,y)= √(x^2+y^2 )

V(x,y)= 0

∂u/∂x=x/√(x^2+y^2 ) ∂u/∂y=y/√(x^2+y^2 )

∂v/∂x=0 ∂v/∂y=0

∴no cumplen con las ecuaciones de Cauchy-Riemann

4.- f(z)=(2z+1)/z

= (2(x+iy)+1)/(x+iy)

= ((2x+2iy+1)/(x+iy))((x-iy)/(x-iy))

= ((2x+2iy+1)(x-iy))/((x+iy)(x-iy))

= (2x^2+2xiy+x-2xiy-2y^2 i^2-iy)/(x^2+y^2 )

= (2x^2+x+2y^2-iy)/(x^2+y^2 )

U(x,y)= (2x^2+x+2y^2)/(x^2+y^2 )

V(x,y)= (-y)/(x^2+y^2 )

∂u/∂x=(y^2-x^2)/〖(x^2+y^2)〗^2 ∂u/∂y=(-2xy)/〖(x^2+y^2)〗^2

∂v/∂x=2xy/〖(x^2+y^2)〗^2 ∂v/∂y=(x^2+y^2)/〖(x^2+y^2)〗^2

∴no satisfacen las ecuaciones de Cauch-Riemann

5.- f(z)=i〖|z|〗^2

= i (√(x^2+y^2 ))^2

= i (x^2+y^2)

U(x,y)= 0

V(x,y)= (x^2+y^2)

∂u/∂x=0 ∂u/∂y=0

∂v/∂x=2x ∂v/∂y=2y

∴no cumplen con las ecuaciones de Cauchy-Riemann

6.- f(z)=z+Im(z)

...