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246A, Notes 2: complex integration

27 September, 2016 in 246A - complex analysis, math.CV, math.GT | Tags: , , | by | 103 comments

Previous set of notes: Notes 1. Next set of notes: Notes 3.
Having discussed differentiation of complex mappings in the preceding notes, we now turn to the integration of complex maps. We first briefly review the situation of integration of (suitably regular) real functions {f: [a,b] \rightarrow {\bf R}} of one variable. Actually there are three closely related concepts of integration that arise in this setting:

  • (i) The signed definite integral {\int_a^b f(x)\ dx}, which is usually interpreted as the Riemann integral (or equivalently, the Darboux integral), which can be defined as the limit (if it exists) of the Riemann sums

    \displaystyle \sum_{j=1}^n f(x_j^*) (x_j - x_{j-1}) \ \ \ \ \ (1)

    where {a = x_0 < x_1 < \dots < x_n = b} is some partition of {[a,b]}, {x_j^*} is an element of the interval {[x_{j-1},x_j]}, and the limit is taken as the maximum mesh size {\max_{1 \leq j \leq n} |x_j - x_{j-1}|} goes to zero (this can be formalised using the concept of a net). It is convenient to adopt the convention that {\int_b^a f(x)\ dx := - \int_a^b f(x)\ dx} for {a < b}; alternatively one can interpret {\int_b^a f(x)\ dx} as the limit of the Riemann sums (1), where now the (reversed) partition {b = x_0 > x_1 > \dots > x_n = a} goes leftwards from {b} to {a}, rather than rightwards from {a} to {b}.

  • (ii) The unsigned definite integral {\int_{[a,b]} f(x)\ dx}, usually interpreted as the Lebesgue integral. The precise definition of this integral is a little complicated (see e.g. this previous post), but roughly speaking the idea is to approximate {f} by simple functions {\sum_{i=1}^n c_i 1_{E_i}} for some coefficients {c_i \in {\bf R}} and sets {E_i \subset [a,b]}, and then approximate the integral {\int_{[a,b]} f(x)\ dx} by the quantities {\sum_{i=1}^n c_i m(E_i)}, where {m(E_i)} is the Lebesgue measure of {E_i}. In contrast to the signed definite integral, no orientation is imposed or used on the underlying domain of integration, which is viewed as an “undirected” set {[a,b]}.
  • (iii) The indefinite integral or antiderivative {\int f(x)\ dx}, defined as any function {F: [a,b] \rightarrow {\bf R}} whose derivative {F'} exists and is equal to {f} on {[a,b]}. Famously, the antiderivative is only defined up to the addition of an arbitrary constant {C}, thus for instance {\int x\ dx = \frac{1}{2} x^2 + C}.

There are some other variants of the above integrals (e.g. the Henstock-Kurzweil integral, discussed for instance in this previous post), which can handle slightly different classes of functions and have slightly different properties than the standard integrals listed here, but we will not need to discuss such alternative integrals in this course (with the exception of some improper and principal value integrals, which we will encounter in later notes).
The above three notions of integration are closely related to each other. For instance, if {f: [a,b] \rightarrow {\bf R}} is a Riemann integrable function, then the signed definite integral and unsigned definite integral coincide (when the former is oriented correctly), thus

\displaystyle \int_a^b f(x)\ dx = \int_{[a,b]} f(x)\ dx

and

\displaystyle \int_b^a f(x)\ dx = -\int_{[a,b]} f(x)\ dx

If {f: [a,b] \rightarrow {\bf R}} is continuous, then by the fundamental theorem of calculus, it possesses an antiderivative {F = \int f(x)\ dx}, which is well defined up to an additive constant {C}, and

\displaystyle \int_c^d f(x)\ dx = F(d) - F(c)

for any {c,d \in [a,b]}, thus for instance {\int_a^b f(x)\ dx = F(b) - F(a)} and {\int_b^a f(x)\ dx = F(a) - F(b)}.
All three of the above integration concepts have analogues in complex analysis. By far the most important notion will be the complex analogue of the signed definite integral, namely the contour integral {\int_\gamma f(z)\ dz}, in which the directed line segment from one real number {a} to another {b} is now replaced by a type of curve in the complex plane known as a contour. The contour integral can be viewed as the special case of the more general line integral {\int_\gamma f(z) dx + g(z) dy}, that is of particular relevance in complex analysis. There are also analogues of the Lebesgue integral, namely the arclength measure integrals {\int_\gamma f(z)\ |dz|} and the area integrals {\int_\Omega f(x+iy)\ dx dy}, but these play only an auxiliary role in the subject. Finally, we still have the notion of an antiderivative {F(z)} (also known as a primitive) of a complex function {f(z)}.
As it turns out, the fundamental theorem of calculus continues to hold in the complex plane: under suitable regularity assumptions on a complex function {f} and a primitive {F} of that function, one has

\displaystyle \int_\gamma f(z)\ dz = F(z_1) - F(z_0)

whenever {\gamma} is a contour from {z_0} to {z_1} that lies in the domain of {f}. In particular, functions {f} that possess a primitive must be conservative in the sense that {\int_\gamma f(z)\ dz = 0} for any closed contour. This property of being conservative is not typical, in that “most” functions {f} will not be conservative. However, there is a remarkable and far-reaching theorem, the Cauchy integral theorem (also known as the Cauchy-Goursat theorem), which asserts that any holomorphic function is conservative, so long as the domain is simply connected (or if one restricts attention to contractible closed contours). We will explore this theorem and several of its consequences in the next set of notes.
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