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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $\\gamma : [a, b] \\to \\mathbb{C}$ be a path that is continuous and piecewise continuously differentiable.\nLet $f$ be a complex function that is continuous on the image $\\gamma([a, b])$.\nThe contour integral of $f$ along $\\gamma$ is\n$$\\int_\\gamma f(z) \\, dz = \\int_a^b f(\\gamma(t)) \\, \\gamma'(t) \\, dt.$$\nThe integral on the right of a complex function $g$ of $t \\in [a, b]$ is $\\int_a^b \\operatorname{Re} g(t) \\, dt + i \\int_a^b \\operatorname{Im} g(t) \\, dt$.\n\n## Remarks\n\nThe length of $\\gamma$ is $L(\\gamma) = \\int_a^b |\\gamma'(t)| \\, dt$.\nThe integral does not change under a reparametrization of $\\gamma$ that keeps the direction. It changes sign if we reverse the direction.\nThe path $\\gamma$ is closed if $\\gamma(a) = \\gamma(b)$. For a closed path, many authors write $\\oint_\\gamma f(z) \\, dz$.\nFor example, along $\\gamma(t) = t + i t$ with $t \\in [0, 1]$, $\\int_\\gamma z \\, dz = \\int_0^1 (1 + i) t (1 + i) \\, dt = i$.\n\n## Explanations\n\n### L1 human\n\nA contour integral adds up the values of $f$ along a path, each one multiplied by a small step $dz$ of the path.\nBecause $dz$ is a complex number, the direction of each step matters, and not only its length.\n",
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