# Extension to all dimensions of complex line integral

Let $$\Gamma$$ be a smooth curve in $$\mathbb{C}^d$$. Since $$\mathbb{C}^d$$ can be seen as $$\mathbb{R}^{2d}$$, one can define the line integral of functions $$f:\Gamma\to \mathbb{C}$$ using for instance arclength parametrization.

However, when $$d=1$$, one can use the usual complex line integral of complex analysis instead. Is there a natural extension of this definition when $$d \geq 2$$ ?

• The most natural extension would have f mapping to $C^d$ rather than $C$. – Michael Renardy Aug 27 at 12:38
• Yes, sure. But i'm dealing with functions defined on curves and the arclength definition of integration does not seem adequate. – Chr Aug 27 at 13:25
• The line integral of complex analysis is the integral of a 1-form, so we can immediately generalize to integration of 1-forms along curves, which is defined in any manifold. The line integral of complex analysis is not an arc length integral of a function. – Ben McKay Aug 27 at 15:06
• Unfortunatlely I'm not familiar with 1-form. Is the function f, defined on $\Gamma$, a 1-form ? – Chr Aug 27 at 16:55
• @Chr: no, a function on a curve is not a 1-form. You can read Spivak, Calculus on Manifolds, for a nice introduction to differential forms. But your question does not appear to be a question of research; perhaps this is the wrong website for your question. – Ben McKay Aug 27 at 20:58