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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 ??
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
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??

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sorry I mean \[w(z)=u(x,y)+j.v(x,y)\]
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 ?
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
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Isn't z a function in it's own that takes 2 reals?? So really a complex function is just a function of a function......
we can think of complex number as a position vector : \((x, y)\) The function machine simply maps the \(xy\) plane into \(uv\) plane |dw:1438098278866:dw|
consider a function that takes reals and outputs reals : \(w = f(x)\) your earlier question is similar to asking : isn't \(x\) another function that takes one real ?
I understand that but what if you can't separate your real and imaginary parts into u and v?? Like in the example \[w=f(z)=\sqrt{z}=\sqrt{x+jy}\] How is something like this mapped into the u,v plane we don't know what is \[u(x,y)\] and \[v(x,y)\]
if you want to separate real and imaginary parts then polar form should work i think but the definition of complex valued function of a complex variable doesn't require us to express the function in terms of components u(x,y) and v(x,y)
Alright, that's what I was thinking the definition is simply of requiring z as an argument

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