Can complex valued functions be written without using x+iy notation?

[Heirot]

Let f(z) be some complex valued function of complex variable z=x+iy. Since f(z) is (in general) complex, we can write it as f(z) = u(z)+iv(z), where u and v are real. But how does one prove that we can also write it as f(z) = u(x,y)+iv(x,y), i.e. shouldn't x and y always appear in the form "x+iy"?

Thanks

 

[nicksauce]

Well u and v are just real valued functions of two real variables. x and y uniquely specify z, therefore we can write u(z) = u(x,y).

 

[Heirot]

ok, but if I have any u(x,y) and v(x,y), can I always find some f(z) so that f(z) = u(x,y) + iv(x,y), where z=x+iy? For example, if I have u(x,y) = x sin(y) and v(x,y) = cos(x), what would f(z) be?

 

[Office_Shredder]

That would be f(z) = Re(z)*sin(Im(z)) + icos(Re(z))

Where if x+iy=z, Re(z):= x, Im(z):= y

 

[lurflurf]

Trivially yes
x=Re(z)
y=Im(z)
so
u(x,y)+iv(x,y)=u(Re(z),Im(z))+iv(Re(z),Im(z))=f(z)
That said
Re(z) and Im(z) are not considered proper functions of z as they break z apart and do not treat it as whole variable
So we are led to the concept of an analystic function that maps z as a whole
The Cauchy-Riemann Equations allow us to check if a function is analystic
for a function in
u+iv form the condition is written
Dx(u)=Dy(v)
Dy(u)=-Dx(v)
where Dx and Dy are partial derivatives with respect to x and y
so for your example
Dx(x sin(y))=Dy(cos(x))
sin(y)=0
Dy(x sin(y))=-Dx(cos(x))
x cos(y)=sin(x)

so we se your function is not analytic and cannot be written as f(z) in a proper way
we could write it as f(z,z*) (where z* is the conjugate of z and z*z=|z|^2)
when written in this form the condition is
Dz*(f)=0 where Dz* is the partial derivative with respect to z*

 

[Heirot]

Yes, that's it! Thank you!