Question

ques

Answer 1

Can we have a complex function in one variable ???
For example consider a complex variable
\[z=x+jy\]
What if we use x instead of y, that is the number with j we r also keeping as x
\[z=x+jx\]
Then a complex function in one variable may be defined
\[w(z)=u(x)+j.v(x)\]
Example
\[\sin(z)=\sin(x+jx)=\sin(x).\cos(jx)+\cos(x).\sin(jx)\]\[\sin(z)=\sin(x).\cosh(x)+j.\cos(x).\sinh(x)\]
Which is of the form
\[\sin(z)=u(x)+j.v(x)\]
where
\[u(x)=\sin(x).\cosh(x)\]\[v(x)=\cos(x).\sinh(x)\]

If I'm not wrong somewhere then why are complex functions started with 2 variables instead of taking an easier approach ??

Answer 2

don't you think the domain of function includes only the complex numbers whose real and imaginary components are equal ? this is as good as a complex valued function that takes real numbers as inputs

Answer 3

So this function is more like a function that only takes numbers from real domain, but by definition a complex function must take complex arguments, the thing's that confusing me is how the function is split after taking an argument, it just seems bizzare
and how would you split something like
\[w(z)=\sqrt{z}=\sqrt{x+jy}\] in the form of \[w(z)+u(x)+j.v(x)\]
Or is it not necessary to split it into that form??

Answer 4

sorry I mean
\[w(z)=u(x,y)+j.v(x,y)\]

Answer 5

its not necessary, how would you write some function like \(w(z) = x^2-y^2 + j (x+x^3y)\) in terms of \(z\) only ?

Answer 6

Oh sorry I read it wrong, the book says it's frequently express as
\[w(z)=u(x,y)+j.v(x,y)\]
and that's not a definition
Makes sense now