VARIABLE known, it is of course a function of them ; but, inasmuch as for real values of x and y, when z( = x + iy} is given, we know both x and y, Cauchy considered any function F(x, y} to be a function of 2. On the other hand, Kiemann con sidered a function F(x, y} to be one of 2 when it contains x and y only in the combination x + iy. In fact, if we have R f Fiji* dF=-sdjc + -yr- dy for any indefinitely small increments OX oy de and dy, inasmuch as this may be written , IS8F .SF. 1/8F .8F, dt = 2 1> + { -fy) ^-^ + 2 Ur * 8y) ( dx + ^> and if dF must always be zero as long as d(x + iy) vanishes, 8F 8F then we must have -s- + i = 0. This, which is the con- 8x Sy dition that, when x and y are perfectly unrestricted (and so may each be complex), they may enter only in the com bination x + iy, must also hold when they can both admit only real values, in order that F may be in Eiemann s sense a function of 2. Cauchy called the function mono- fjene when it satisfies this relation. The further signifi cance of this relation is seen if we put dx = p cos$ and dy = p siu#, whence da = pe i9 and dx idy = pe~ i9 ; for then in general 1 /8F ~ . 8F adjacent point. When - + 1-5- = we have -7- = -5- = -r -*- : 8x b dz ox i oy o v s " T * s- /<"**+ o( "5" -* TT ) depends
- 2 or oy / 2 oj? oy /
not only on the value of z but also on the direction of the 8F 8F dF 8F 1 8F ,-K- = Owe have -j- by dz ox i that is, for each point 2, in case the function is monoyeneous, there is a derived function with regard to the complex variable, independent of the direction in which this vari able is supposed to change. If the relation do not hold, there is a different derived for each such direction. When we separate F into its real and imaginary parts, we may write F = u + iv, and the preceding relation becomes 811, 8v ./Sy 8ii 8u 8v 8u 8t> s 2~ + M"s~ + F~)=0, whence s ^- = and 5- + -5- = 0. bx by bx byj ox 8y Sy ox s 2 s>2 j2 K2 , , O H O U . O V V Hence further T~T, + *-g = and ^ + ^-r, = 0. ox- by* bx- by- 9. A function of a complex variable which is continuous, one-valued, and has a derived function when the variable moves in a certain region of the plane is called by Cauchy si/nectic in this region, and by Briot and Bouquet l hofa- iiiorphe, to indicate that it is like an integer function, for which this property holds throughout the entire plane. Weierstrass styles it an analytic function. The property that dw/dz is independent of dy/dx furnishes a geometrical relation between the figure which w describes in its plane and that which 2 describes. In fact, let values ?^, ?/>. ?o> correspond to the values 2, 9 , 2,, then, if - :! = - -, 1 O 1 ni nit y /- "2 ( "i ~2 ~ ~i as already observed, the triangle for the values iv is similar and similarly placed to that for the values 2 ; in that case yi) 7/f f>n ijj . But, if we suppose z 3 to close up to Z L , .:., z l z. 2 - 2 X then u 3 closes up to w^ and, if z. 2 close up to 2 1; w., closes up to u. In the limit the equality of these quotients for all directions of variation of z shows that the figure which ic describes in its plane is in its infinitely small parts similar to that described by 2, and two intersecting curves 1 The word monodrome is used by Cauchy ami monotrope by Briot .rid Bouquet to indicate the case in which, supposing z has to move in a certain region, all the paths that lead from an initial point z a to any point z within the region lead to one and the same value of ?" ; in other cases w will be poly trope. When w is holomnrphe in a region except at a point -j, where it becomes infinite without- 1/w ceasing to be holomorphe near this, this point is called a, pole or an infinity of w. A rational fraction has the roots of its denominator as poles ; and a Junction otherwise holomorphe having poles is called by these writers t.)- romnrpbe. Some writers prefer the terms integer and fractional functions. in the plane of w cut at the same angle as the correspond ing curves in the plane of 2, provided always that dw dz is neither zero nor infinity. 10. Let 2 + k be a complex value within the circle of con vergence of the series/(Y) = + a^z + </ 2 2- + . . . + a a z n + . . . , then we have f(z + h) = + a^z + Ji) + a. 2 (z + A) 2 + . . . + a n (z + //)" + . . . Now, if we arrange this absolutely convergent series by powers of h, the coefficients are infinite series which can be proved to be the successive derived functions of f(z) ; thus, when a one- valued function /() can be expanded in a series of powers of 2, the expansion can be made only in a single manner. We have therefore /( " which is Taylor s theorem for a function given by an in finite series of powers of a complex variable. 11. A r e proceed to examine at how many points within a circle of convergence of radius 7? the function /(:) vanishes, i.e., has a root. In the first place it can be shown that, if R be finite, the function cannot vanish at infinitely many points in the circle without vanishing- identical Iy, that is, being zero everywhere in the circle. Hence we can only have a finite number of roots in an}" finite portion of the plane, and we may suppose none of these to lie upon the boundary circle. If the function vanish at any point 2 1? then, when we take the expansion f(z) -fo + , - O -fa) + - k) + ^i) + -fee., we have f(z^) = 0. We may also have f (z^) = 0, /"(^ ) = 0, itc. Suppose that the first of these which does not vanish is/( a )(2 1 ), or that the function /() vanishes for 2= .~ l to the order a, then -ja = |1 /< a) (-*i) + ^ / (a+1) 0i) + ^ Now let this quotient be rearranged according to powers of 2, and suppose the function to vanish at 2., to the order f3, we can expand by powers of z - z.,, whence we get- ^ , - , ultimately =C/,(.E), a series not (Z 2J (2 Z. 2 ) vanishing anywhere within the region. If /(.:) were a ra tional integer algebraic function, we should find </>(:) a constant, and we see that a + (3 + . . . <.tc. = n : that is, such a function cannot have more than n roots in the region of convergence, i.e., in the entire plane. Taking logarithms, we have log/(.?) = alog( - 2^ + jklogfc - z. 2 ) + . . . + log ^(2). Hence, taking for each logarithm on the right one of its innumerable values, let 2 move from any point A round the circumference of the boundary circle, keeping its interior on the left. Since 2 - z i vanishes only once within the circle, namely, for 2 = 2 1? log(2 - 2 a ) = log /a + iO will become log/o + t(# + 27r) when z returns to A. Similarly log(2-2 2 ) will differ from its initial value by 2/~; and so for the others. But, as $(z) does not vanish anywhere in the circle, log $(2) will return to its original value at A. Hence, if there be v vanishing points or roots within the circle of convergence, the value of log/(.r) changes by lirrv, when 2 travels all round the circumference. Conversely, if the logarithm alter by li-n-v as 2 travels round the circle of convergence, the number of roots within the circle is i . 12. This theorem of Cauchy s at once leads to the fundamental theorem of algebra, that every equation of the wth degree /(.--) = a + az + a.,i- + . . . + a,,,: 1 = has n complex roots. For we can take the radius of the circle so great that the value of a lt z n far exceeds that of the other terms; and, writing f(z) a n z n ( + P}, the modulus of J" can be made as small as we please. Hence log f(z) = log(rt M 2 w ) + log(l + /*). Now as we go round the circle log(l + /*) differs as little as we please from log(l) = 2/-?V,
and so, whatever value we begin with, as the corresponding