4.1 Discussion

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Since we have discussed di_erentiating complex functions, it is now natural

to turn to the problem of integrating them.

Brooding on what it might mean to integrate a function f : C 􀀀! C we

might conclude that there are two factors which need to be considered.

The _rst is that integration ought to still be a one-sided inverse to di_erentiation;

di_erentiating an inde_nite integral of a complex function should

yield the function back again. The second is that integration ought still to

be something to do with adding up numbers associated with little boxes and

taking limits as the boxes get smaller.

We have just been discussing writing out a vector _eld as the conjugate of a

complex function, so there is a good prospect that we can integrate complex

functions over curves, by thinking of them as vector _elds. In second year you

managed to make sense of integrating vector _elds over curves and surfaces,

and should now feel cheerful about doing this in the plane. So your experience

of integration already extends to two and three dimensions, and you recall,

I hope, the planar form of the Fundamental Theorem of Calculus known as

Green's Theorem. If you don't, look it up in your notes, you're going to need

it.

On the other hand, we could just take the real and imaginary parts separately,

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106 CHAPTER 4. INTEGRATION

and integrate each of these in the usual way as a function of two variables.

This would give us some sort of complex number associated with a function

and a region in C . If we were to try to 'integrate' the function 2z in this way,

to get an inde_nite integral, we would get x2y + iy2x, which is not complex

di_erentiable except at the origin. If the FTC is to hold, di_erentiating an

inde_nite integral ought to get us back to the thing integrated, and here it

does no such thing. So we conclude that this is not a particularly useful way

to de_ne a complex integral.

Now the derivative of a complex function is a complex function, so the integral

of a complex function should also be a complex function. So integrating

functions from C to C to get other functions from C to C must be more

like integrating functions from R to R than integrating or vector _elds. This

leads to the issue: what do we integrate over? If we integrate over regions

in C , then any version of the Fundamental Theorem of Calculus has to be

some variant of Green's Theorem, and must be concerned with relating the

integral over the region of one function with the integral over the boundary

of another. So we seem to need to integrate complex functions over curves if

we need to integrate them over regions. And we know how to integrate along

curves, because a complex function f(z) is a vector _eld in an obvious way.

Another argument for thinking that curves are the things to integrate complex

functions over is that if we have an expression like

Z f(z)dz

then the dz ought surely to be dx+i dy and this is an in_nitesimal complex

number, representable perhaps as a very, very small arrow. And not as a

very, very small square.

Intuitive arguments of this sort can merely be suggestive, since they are

derived from our experience on a di_erent world, the world of real functions.

There is a school of thought which would ban such arguments on the grounds

that they can lead us astray, but it is more useful to go somewhere on the

strength of a risky analogy than to go nowhere because it is safer. Anyway,

it isn't.

We therefore investigate to see if integrating a complex function along a curve

is generally a reasonable thing to do.

4.2. THE COMPLEX INTEGRAL 107

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