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transition from the plane of z to that of w, all the infinitesimal vectors drawn from the point P have their lengths altered in the same ratio, and are turned through the same angle. Hence any angle in the plane of z is equal to the corresponding angle in the plane of w, and any infinitely small figure in the one plane is similar to the corresponding figure in the other. In other words, corresponding figures in the planes of z and w are similar in their infinitely small parts.

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For instance, in the plane of w the straight lines const., const., where the constants have assigned to them a series of values in arithmetical progression, the common difference being infinitesimal and the same in each case, form two systems of straight lines at right angles dividing the plane into infinitely small squares. Hence in the plane xy the corresponding curves const., const., the values of the constants being assigned to them as before cut one another at right angles (as has already been proved otherwise) and divide the plane into a series of infinitely small squares.

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The similarity of corresponding infinitely small portions of the planes w and z breaks down at points where the differential coefficient

dw
dz

is zero or infinite. Since

= + i

dy

the

dw do
dz dx dx'

corresponding value of the velocity, in the hydrodynamical application, is zero or infinity.

77. The processes of differentiation and integration, as applied to functions of a complex variable, claim a little notice.

The conditions (13) that w should be a function of z may be written

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a form which the student should interpret. It is obvious that (14)

is satisfied when

dw

dz

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is itself a function of z. Hence all the derivatives of a function

of a complex variable are themselves functions of that variable.

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taken along any assigned path from z, to z, is defined as follows. Supposing the path divided into infinitesimal portions, we form the product wdz for each of them, and add the results. It is easily shewn that the value of the integral is to a certain extent independent of the nature of the path joining z and 2. The theorem of Art. 40, when applied to a plane space (xy) becomes

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The double integral extends over any portion of the plane xy throughout which u and v are finite and continuous, and their first derivatives finite; whilst the single integral extends in the positive direction (see Art. 39) round the boundary of that portion. Now if we write u=w, v=iw, it follows by (14) that s(wdx +iwdy), or fwdz, is zero when taken round the boundary of any portion of the plane xy throughout which w is finite and continuous, and its first derivative finite.

We infer, as in Art. 41, that the integral (15) is the same for any two paths joining z, z,, so long as these paths do not include between them any points at which w is infinite or discontinuous,

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Any points at which these conditions are violated may be isolated by drawing an infinitely small closed curve around each. The rest of the plane xy then forms a multiply-connected region. The value of the integral fwdź taken round any evanescible circuit drawn in this region is zero; and the integral (15) is the same for any two reconcileable paths. The values of (15) corresponding to two irreconcileable paths differ by a quantity of the form P12+P22 + ......, where P1, P2...... are integers, and 1, *,,...... denote the values of fwdz taken round the several circuits surrounding the above-mentioned points*. It is unnecessary to

* In the analytical theory K1, K2,... are called the 'moduli of periodicity' of the integral (15).

dwell on the proof of these statements, or to enter more fully into the theory of many-valued integrals of the form (15); to do so would be to repeat, with merely verbal alterations, portions of Chapter III.

The integral (15) is itself a function of z, according to the definition of Art. 74. For, denoting its value by Z, we have,

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78. An important illustration of the above theory is furnished by the integral

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The only point at which the function z", or its derivative, is infinite or discontinuous, is the origin. Introducing polar coordinates, we write

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Hence the value of (17) taken round an infinitely small circle having the origin as centre is

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In the analytical theory above referred to, the logarithm of a complex quantity z is defined by the equation

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Hence log z is a many-valued function, the cyclic constant being 2πi.

The properties of the logarithmic function readily follow from the definition (20). Thus if z,, z, be any two complex quantities we have

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or rather, since it appears that the real part of the logarithm of a small quantity is essentially negative,

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let us investigate the properties of z as a function of w. This function we denote by ew. It follows at once from (21) that

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the fundamental property of the 'exponential' function. Hence, and from (22), (23),

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Also since log z is cyclic, the constant being 2πi, e" is a periodic function, the period being 2πi, viz.

where n is any integer.

ew±2nπi = ew,

Let us map out, in the manner explained in Art. 76, the relation between the two functions z and w. It appears from (19)

that

w=log z =

rr dr
+ io......

r

.(25)*.

The first term on the right-hand side of (25) is essentially real; we denote it by logr. We have then logr and as the rect

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angular co-ordinates of the point P' in the plane of w corresponding If P describe a circle of

to the point P (x, y) of the plane of z.

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radius unity about the origin, starting from the point (1, 0); then, since log 10, P' will move along the axis of , and will describe a length 27 of that axis for every revolution of P. To points P outside the circle r = 1 correspond points P' to the right of the axis of; and vice versa. To every straight line through the origin, in the plane of z, corresponds in the plane of w a straight line parallel to the axis of p. The periodicity of z, and the cyclosis of w, are manifested by the division of the plane of w by the straight lines ↓ = 2nπ into an infinite number of compartments, each of which corresponds to the whole of the plane of z.

In the same way the properties of the function arc tan z may be deduced from the definition

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and, though with much greater difficulty, those of the function arc sin z from the definition

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