In what precedes we have used ? to denote the

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In what precedes we have used ζ to denote the argument of the func- tion Log ζ , and ( ξ, η ) or ( ρ, φ ) to denote the coordinates of ζ ; and z , ( x, y ), ( r, θ ) to denote an arbitrary point on the path of integration and its coor- dinates. There is however no reason now why we should not revert to the natural notation in which z is used as the argument of the function Log z , and we shall do this in the following examples. Examples XCIII. 1. We supposed above that - π < θ < π , and so excluded the case in which z is real and negative . In this case the straight line from 1 to z passes through 0, and is therefore not admissible as a path of integration. Both π and - π are values of am z , and θ is equal to one or other of them: also r = - z . The values of Log z are still the values of log | z | + i am z , viz. log( - z ) + (2 k + 1) πi, where k is an integer. The values log( - z ) + πi and log( - z ) - πi correspond to paths from 1 to z lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of Log z , as convenience dictates. We shall choose the value log( - z )+ π i corresponding to the first path. 2. The real and imaginary parts of any value of Log z are both continuous functions of x and y , except for x = 0, y = 0. 3. The functional equation satisfied by Log z . The function Log z satisfies the equation Log z 1 z 2 = Log z 1 + Log z 2 , (1) in the sense that every value of either side of this equation is one of the values of the other side. This follows at once by putting z 1 = r 1 (cos θ 1 + i sin θ 1 ) , z 2 = r 2 (cos θ 2 + i sin θ 2 ) , and applying the formula of p. 503 . It is however not true that log z 1 z 2 = log z 1 + log z 2 (2)
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[X : 221] THE GENERAL THEORY OF THE LOGARITHMIC, 504 in all circumstances. If, e.g. , z 1 = z 2 = 1 2 ( - 1 + i 3) = cos 2 3 π + i sin 2 3 π, then log z 1 = log z 2 = 2 3 πi , and log z 1 + log z 2 = 4 3 πi , which is one of the values of Log z 1 z 2 , but not the principal value. In fact log z 1 z 2 = - 2 3 πi . An equation such as (1), in which every value of either side is a value of the other, we shall call a complete equation, or an equation which is completely true . 4. The equation Log z m = m Log z , where m is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true. 5. The equation Log(1 /z ) = - Log z is completely true. It is also true that log(1 /z ) = - log z , except when z is real and negative. 6. The equation log z - a z - b = log( z - a ) - log( z - b ) is true if z lies outside the region bounded by the line joining the points z = a , z = b , and lines through these points parallel to OX and extending to infinity in the negative direction. 7. The equation log a - z b - z = log 1 - a z - log 1 - b z is true if z lies outside the triangle formed by the three points O , a , b . 8. Draw the graph of the function I (Log x ) of the real variable x . [The graph consists of the positive halves of the lines y = 2 and the negative halves of the lines y = (2 k + 1) π .] 9. The function f ( x ) of the real variable x , defined by πf ( x ) = + ( q - p ) I (log x ) , is equal to p when x is positive and to q when x is negative.
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