Argument Principle - Proof of The Argument Principle

Proof of The Argument Principle

Let zN be a zero of f. We can write f(z) = (zzN)kg(z) where k is the multiplicity of the zero, and thus g(zN) ≠ 0. We get

and

Since g(zN) ≠ 0, it follows that g' (z)/g(z) has no singularities at zN, and thus is analytic at zN, which implies that the residue of f′(z)/f(z) at zN is k.

Let zP be a pole of f. We can write f(z) = (zzP)−mh(z) where m is the order of the pole, and thus h(zP) ≠ 0. Then,

and

similarly as above. It follows that h′(z)/h(z) has no singularities at zP since h(zP) ≠ 0 and thus it is analytic at zP. We find that the residue of f′(z)/f(z) at zP is −m.

Putting these together, each zero zN of multiplicity k of f creates a simple pole for f′(z)/f(z) with the residue being k, and each pole zP of order m of f creates a simple pole for f′(z)/f(z) with the residue being −m. (Here, by a simple pole we mean a pole of order one.) In addition, it can be shown that f′(z)/f(z) has no other poles, and so no other residues.

By the residue theorem we have that the integral about C is the product of 2πi and the sum of the residues. Together, the sum of the k 's for each zero zN is the number of zeros counting multiplicities of the zeros, and likewise for the poles, and so we have our result.

Read more about this topic:  Argument Principle

Famous quotes containing the words proof of, proof, argument and/or principle:

    Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?
    Henry David Thoreau (1817–1862)

    The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.
    Andrew Michael Ramsay (1686–1743)

    “English! they are barbarians; they don’t believe in the great God.” I told him, “Excuse me, Sir. We do believe in God, and in Jesus Christ too.” “Um,” says he, “and in the Pope?” “No.” “And why?” This was a puzzling question in these circumstances.... I thought I would try a method of my own, and very gravely replied, “Because we are too far off.” A very new argument against the universal infallibility of the Pope.
    James Boswell (1740–1795)

    The principle of fashion is ... the principle of the kaleidoscope. A new year can only bring us a new combination of the same elements; and about once in so often we go back and begin again.
    Katharine Fullerton Gerould (1879–1944)