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Home | 18.013A | Chapter 20 |
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The definite integral considered so far represents area in the plane. We defined it however by dividing the interval in into small subintervals (say of width ) and taking the sum of an estimate of the area of each subinterval, namely where is a point in that subinterval.
Suppose now we let be a curve in the complex plane, and let be a function of the variable with . We can define integration along that curve of an integrand by dividing the curve into small pieces, and summing where is a point in the i-th interval whose endpoints are and , over all the pieces.
This integral along a curve will no longer represent area, since neither nor the difference between 's will be real numbers. But we can multiply complex numbers together, so that this definition makes perfect sense.
Such an entity is called a contour integral in the complex plane. Though this integral no longer has the interpretation of area, it still has the property that it is well defined if the path is finite and the function is bounded and continuous on it. We denote it as follows
Integrals of this kind are enormously valuable mathematical tools, as we may soon see.
Do not be put off by the strangeness of integrating in the complex plane and dealing with functions that have complex values. Almost everything you can say about ordinary integrals apply to these.
Exercises:
20.3 Integrate the function up the imaginary axis, from 0 to using what you know about the integral of the sine function.
20.4 Do the same thing numerically on a spreadsheet, writing
and
, with
(and hence
),
going from 0 to 1.
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