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**Additional info for Elements of the Theory of Functions of a Complex Variable**

**Example text**

MULTIFORM FUNCTIONS. 47 round each branch-point and let the variable describe the same in succession, then this path can be deformed, without line passing over one of the branchpoints, into a closed line which, starting from branch-points. a and z , encloses all the where two branch- (Fig. ) lines round the individual branchpoints most simply, by describing round each one a small circle, and connecting each of these circles with must then be described 10. We twice, going will next illustrate by some examples, and at the % by a and coming.

MULTIFORM FUNCTIONS. The introduction of complex variables also throws a on the nature of multiform (many-valued) functions. For, since a complex variable may. describe very different paths in passing from an initial point %Q to another point the question naturally suggests itself, whether the path described cannot affect the value w which a function, starting with a definite value w corresponding to ZQ) acquires at the terminal point %; we have to inquire whether the curves described by w, starting from w which correspond to those described between Z Q and z 1} must always end in the same point WD or whether they cannot also end in different points.

And then by equating real and imaginary Finally, we can we obtain parts, establish for each of these functions a single of condition. Eor, differentiating each of the above equation x and y, and eliminating v and u in to as equations partially turn, so that neither of the functions u and v is arbitrary, but each one must satisfy the same partial differential equation. As is well known, partial differential equations do not characterize particular functions but general classes of functions. Thus the function w tion (1), and the (2) and of the complex variable z = x + iy is given by equar real constituent parts of such a function by (3).