Abstract
Complex analysis, just like real analysis, deals with questions of continuity, convergence of series, differentiation, integration, and so forth. The reader is assumed to have been exposed to the algebra of complex numbers.
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Notes
- 1.
Strictly speaking, we should write \(f:S\to\mathbb{C}\) where S is a subset of the complex plane. The reason is that most functions are not defined for the entire set of complex numbers, so that the domain of such functions is not necessarily \(\mathbb{C}\). We shall specify the domain only when it is absolutely necessary. Otherwise, we use the generic notation \(f:\mathbb{C}\to\mathbb{C}\), even though f is defined only on a subset of \(\mathbb{C}\).
- 2.
One can rephrase this and say that the derivative exists, but not in terms of ordinary functions, rather, in terms of generalized functions—in this case θ(x)—discussed in Sect. 7.3.
- 3.
We use z ∗ to indicate the complex conjugate of z. Occasionally we may use \(\bar{z}\).
- 4.
Although the derivative of |z|2 exists at z=0, it is not analytic there (or anywhere else). To be analytic at a point, a function must have derivatives at all points in some neighborhood of the given point.
- 5.
We use electrostatics because it is more familiar to physics students. Engineering students are familiar with steady state heat transfer as well, which also involves Laplace’s equation, and therefore is amenable to this technique.
- 6.
This statement is valid only in Cartesian coordinates. But these are precisely the coordinates we are using in this discussion.
- 7.
We are using z′ instead of w, and (x′,y′) instead of (u,v).
- 8.
Writing z=|z|e iθ, we note that lnz=ln|z|+iθ, so that the real part of a complex log function is the log of the absolute value.
- 9.
The interchange of differentiation and integration requires justification. Such an interchange can be done if the integral has some restrictive properties. We shall not concern ourselves with such details. In fact, one can achieve the same result by using the definition of derivatives and the usual properties of integrals.
- 10.
Note that the integral is path-independent due to the analyticity of f. Thus, F is well-defined.
- 11.
As remarked before, the series diverges for all points outside the circle |z|=1. This does not mean that the function cannot be represented by a series for points outside the circle. On the contrary, we shall see shortly that Laurent series, with negative powers of z−z 0 are designed precisely for such a purpose.
- 12.
This is a reflection of the fact that the function is not analytic inside the entire circle |z|=1; it blows up at z=0.
- 13.
We could, of course, evaluate the derivatives of all orders of the function at z=0 and use Maclaurin’s formula. However, the present method gives the same result much more quickly.
References
Churchill, R., Verhey, R.: Complex Variables and Applications, 3rd edn. McGraw-Hill, New York (1974)
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Hassani, S. (2013). Complex Calculus. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_10
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