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4.6: Cauchy's Theorem

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    6485
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    Cauchy’s theorem is analogous to Green’s theorem for curl free vector fields.

    Theorem \(\PageIndex{1}\) Cauchy's theorem

    Suppose \(A\) is a simply connected region, \(f(z)\) is analytic on \(A\) and \(C\) is a simple closed curve in \(A\). Then the following three things hold:

    (i) \(\int_{C} f(z)\ dz = 0\)

    (i') We can drop the requirement that \(C\) is simple in part (i).

    (ii) Integrals of \(f\) on paths within \(A\) are path independent. That is, two paths with the same endpoints integrate to the same value.

    (iii) \(f\) has an antiderivative in \(A\).

    Proof

    We will prove (i) using Green’s theorem – we could give a proof that didn’t rely on Green’s, but it would be quite similar in flavor to the proof of Green’s theorem.

    Let \(R\) be the region inside the curve. And write \(f = u + iv\). Now we write out the integral as follows

    \[\int_{C} f(z)\ dz = \int_{C} (u + iv) (dx + idy) = \int_{C} (u\ dx - v\ dy) + i(v \ dx + u\ dy). \nonumber \]

    Let’s apply Green’s theorem to the real and imaginary pieces separately. First the real piece:

    \[\int_{C} u \ dx - v\ dy = \int_{R} (-v_x - u_y) \ dx\ dy = 0. \nonumber \]

    Likewise for the imaginary piece:

    \[\int_{C} v\ dx + u\ dy = \int_R (u_x - v_y) \ dx\ dy = 0. \nonumber \]

    We get 0 because the Cauchy-Riemann equations say \(u_x = v_y\), so \(u_x - v_y = 0\).

    To see part (i′) you should draw a few curves that intersect themselves and convince yourself that they can be broken into a sum of simple closed curves. Thus, (i′) follows from (i). (In order to truly prove part (i′) we would need a more technically precise definition of simply connected so we could say that all closed curves within \(A\) can be continuously deformed to each other.)

    Part (ii) follows from (i) and Theorem 4.4.2.

    To see (iii), pick a base point \(z_0 \in A\) and let

    \[F(z) = \int_{z_0}^{z} f(w)\ dw. \nonumber \]

    Here the itnegral is over any path in \(A\) connecting \(z_0\) to \(z\). By part (ii), \(F(z)\) is well defined. If we can show that \(F'(z) = f(z)\) then we’ll be done. Doing this amounts to managing the notation to apply the fundamental theorem of calculus and the Cauchy-Riemann equations. So, let’s write

    \[f(z) = u(x, y) + iv (x, y),\ \ \ \ \ \ F(z) = U(x, y) + iV (x, y). \nonumber \]

    Then we can write

    \[\dfrac{\partial f}{\partial x} = u_x + iv_x, \text{etc.} \nonumber \]

    We can formulate the Cauchy-Riemann equations for \(F(z)\) as

    \[F'(z) = \dfrac{\partial F}{\partial x} = \dfrac{1}{i} \dfrac{\partial F}{\partial y} \nonumber \]

    i.e.

    \[F'(z) = U_x + iV_x = \dfrac{1}{i} (U_y + i V_y) = V_y - i U_y. \nonumber \]

    For reference, we note that using the path \(\gamma (t) = x(t) + iy (t)\), with \(\gamma (0) = z_0\) and \(\gamma (b) = z\) we have

    \[\begin{array} {rcl} {F(z) = \int_{z_0}^{z} f(w)\ dw} & = & {\int_{z_0}^{z} (u (x, y) + iv(x, y)) (dx + idy)} \\ {} & = & {\int_0^b (u(x(t), y(t)) + iv (x(t), y(t)) (x'(t) + iy'(t))\ dt.} \end{array} \nonumber \]

    Our goal now is to prove that the Cauchy-Riemann equations given in Equation 4.6.9 hold for \(F(z)\). The figure below shows an arbitrary path from \(z_0\) to \(z\), which can be used to compute \(f(z)\). To compute the partials of \(F\) we’ll need the straight lines that continue \(C\) to \(z + h\) or \(z + ih\).

    屏幕快照 2020-09-06 下午5.39.03.png
    Paths for proof of Cauchy’s theorem

    To prepare the rest of the argument we remind you that the fundamental theorem of calculus implies

    \[\lim_{h \to 0} \dfrac{\int_0^h g(t)\ dt}{h} = g(0). \nonumber \]

    (That is, the derivative of the integral is the original function.)

    First we'll look at \(\dfrac{\partial F}{\partial x}\). So, fix \(z = x + iy\). Looking at the paths in the figure above we have

    \[F(z + h) - F(z) = \int_{C + C_x} f(w)\ dw - \int_C f(w) \ dw = \int_{C_x} f(w)\ dw. \nonumber \]

    The curve \(C_x\) is parametrized by \(\gamma (t) + x + t + iy\), with \(0 \le t \le h\). So,

    \[\begin{array} {rcl} {\dfrac{\partial F} {\partial x} = \lim_{h \to 0} \dfrac{F(z + h) - F(z)}{h}} & = & {\lim_{h \to 0} \dfrac{\int_{C_x} f(w)\ dw}{h}} \\ {} & = & {\lim_{h \to 0} \dfrac{\int_{0}^{h} u(x + t, y) + iv(x + t, y)\ dt}{h}} \\ {} & = & {u(x, y) + iv(x, y)} \\ {} & = & {f(z).} \end{array} \nonumber \]

    The second to last equality follows from Equation 4.6.10.

    Similarly, we get (remember: \(w = z + it\), so \(dw = i\ dt\))

    \[\begin{array} {rcl} {\dfrac{1}{i} \dfrac{\partial F}{\partial y} = \lim_{h \to 0} \dfrac{F(z + ih) - F(z)}{ih}} & = & {\lim_{h \to 0} \dfrac{\int_{C_y} f(w) \ dw}{ih}} \\ {} & = & {\lim_{h \to 0} \dfrac{\int_{0}^{h} u(x, y + t) + iv (x, y + t) i \ dt}{ih}} \\ {} & = & {u(x, y) + iv(x, y)} \\ {} & = & {f(z).} \end{array} \nonumber \]

    Together Equations 4.6.12 and 4.6.13 show

    \[f(z) = \dfrac{\partial F}{\partial x} = \dfrac{1}{i} \dfrac{\partial F}{\partial y} \nonumber \]

    By Equation 4.6.7 we have shown that \(F\) is analytic and \(F' = f\).


    This page titled 4.6: Cauchy's Theorem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.