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Dragonfall
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I'm trying to learn some elementary complex variables, and I was reading this book on it when I came upon this
Consider the function [tex]f(x)=\left\{\begin{array}{cc}0\Leftrightarrow Im(x)\neq 0\\e^x\Leftrightarrow Im(x)=0\end{array}\right[/tex]. Wouldn't it make more sense if we had a concept of "directional infinity"? If there were only 1 point at infinity on the complex plane, does f converge at infinity?
In dealing with complex numbers we also speak of infinity, which we call "the complex number infinity." It is designated by the usual symbol. We do not give a sign to the complex infinity nor define its argument. Its modulus, however, is larger than any preassigned real number.
We can imagine that the complex number infinity is represented graphically by a point in the Argand plane[...]
Consider the function [tex]f(x)=\left\{\begin{array}{cc}0\Leftrightarrow Im(x)\neq 0\\e^x\Leftrightarrow Im(x)=0\end{array}\right[/tex]. Wouldn't it make more sense if we had a concept of "directional infinity"? If there were only 1 point at infinity on the complex plane, does f converge at infinity?