calculus and analysis - Definite Integral over a path

What does

Integrate[f[z], {z, a, b, c, d}]

exactly calculate? Is it

$$\int_a^b f(z)\, \mathrm{d}z +\int_b^c f(z)\, \mathrm{d}z +\int_{c}^d f(z)\, \mathrm{d}z ?$$

That was my first idea but

Integrate[f[z], {z, a, b, a}]

isn't simplified to 0 (even with a FullSimplify).

For example

   Integrate[1/z, {z, 1, I, -1, -I, 1}]

gives as result $2\cdot \pi \cdot i$, which is the value of any closed path around zero.

Which path does it take in the complex plane? Because there has been an Integral, where I thought it would converge, but I got an error with this one.

Answer

This syntax is correct and it does what you were guessing. However, the simplification is done only if you enter a specific function:

Integrate[x^2, {x, a, b, c, a}]

0

In other words, I replaced your symbolic f by x^2. I think it is reasonable to not simplify such integrals unless you know that the path of the integration doesn't enclose (or in 1D, touch) any singularities. But this can't be guaranteed if the function is unspecified.

Edit

The segments between the points in the specification a,b,c... are straight lines if the integral is more than one-dimensional. In case you're interested in more general line integrals (either in the complex plane or otherwise), here ia a related post:

Complex line integral

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