Elementary Functions of a Complex Variable

We now proceed to define functions such as , , , and so on, taking care that these definitions reduce to the conventional ones when becomes the real variable .

The simplest such function is the integral power function

where is a positive integer or zero. This function is clearly defined by repeated multiplication according to the law

A polynomial is then defined as a linear combination of a finite number of such functions, where the constants of combination may he complex,

A rational function of is defined as the ratio of two polynomials.

By considering the limit of expressions of form (2.3) as or, more generally, limits of the form

where may be real or complex, we are led to definitions of functions of a complex variable in terms of power series. The convergence of such series may be investigated by the ratio test red{(see the chapter on series in a calculus book)}, just as in the case of series of real terms. Thus, if we write

when that limit exists, the series converges when

Geometrically, this restriction is seen to require that lie inside a circle of radius with center at the point in the complex plane. Inside this circle of convergence the series can be integrated or differentiated term by term, and the resultant series will represent the integral or derivative, respectively, of the represented function.

The exponential function is defined by the power series

This definition is acceptable, since the series converges and hence defines a function of for all real or complex values of , and since this series reduces to the usual definition when is real. If we multiply the series defining and together term by term (as is permissible for convergent power series) the resultant series is found to be that defining ; that is, the relation

is true for complex values of and . Consequently, if is a positive integer, we have also the relation

for all complex values of .

The circular functions may be defined by the relations

together with the relations , and so on. Consequently, we have, from (2.10) and (2.7), the corresponding series definitions

and

which reduce to the proper forms when is real. From these series, or from the definitions and (2.10), it can be shown that the circular functions satisfy the same identities for complex values of as for real values. Equations and (2.10) imply the relation

which is known as Euler's formula.

With this relation equation (1.9) takes the form

In consequence of the (2.13) and (2.9) we then have

But since (2.12) also implies the relation

we deduce DeMoirre's theorem,

and hence may rewrite (2.14) in the form

Geometrically, (2.16) shows that if has the absolute value and the angle , then has the absolute value and the angle , if is a positive integer.

In a similar way we find that if and , then

and

That is, if and have absolute values and and angles and , respectively, then has the absolute value and the angle , and has the absolute value and the angle . In particular, we notice that since

the multiplication of any complex number by a number of the form , where is real, is equivalent to rotating the vector representing the number through an angle in the complex plane.

The hyperbolic functions are defined, as for functions of a real variable, by the equations

and by the equations , and so on. Consequently, we have also

From these definitions it can be shown that hyperbolic functions of a complex variable satisfy the same identities as the corresponding functions of a real variable. By comparing and (2.10) with and (2.19) for circular and hyperbolic functions, we obtain in particular the relations

relating the circular and hyperbolic functions.

The results so far obtained permit us to express the functions considered in terms of their real and imaginary parts as follows:

We next define the complex logarithmic function as the inverse of the exponential function. Denoting this function temporarily by , the equation

must then be equivalent to

To express in terms of its real and imaginary parts, we write

after which (2.24) gives

Hence, equating real and imaginary parts, we obtain

Solving for and , there follows

and hence

where represents, as usual, the absolute value of and is the ordinary real logarithm, whereas is a particular choice of the infinitely many angles (differing by integral multiples of ) which may be associated with . Thus we have obtained the result

To emphasize the fact that is determinate only within an integral multiple of , we may here denote that particular value of which lies in the range by , and may speak of this value as the principal value of for the logarithm,

Then any other permissible value of is of the form , when is integral, and (2.28) becomes

Thus it follows that the function , defined as the inverse of , is an infinitely many-valued function. For example, if , there follows and , and hence

The value corresponding to is frequently called the principal value of the logarithm.

If, in a particular discussion, is restricted to real positive values, say , there follows and , and hence, from (2.30),

Thus the complex logarithm of a positive real number may differ from the usual real logarithm by an arbitrary integral multiple of . In order to conform with conventional usage, henceforth we will identify the complex logarithm with the real logarithm when, throughout a given discussion, is real and positive, by taking in (2.32) in such a case. In the more general case it is also conventional to write in place of , with the understanding that unless is to take on only real positive values, is to be considered as multiply valued. Thus we will write

in place of (2.30), and avoid resultant contradiction when is a positive real variable by taking .

Suppose now that for a given point in the complex plane we choose a particular value of in (2.33), say , and hence determine a particular value of . If a point moves continuously along a path originating at , the value of then varies continuously from the initial value . In particular, if traverses a closed path surrounding the origin in the positive (counterclockwise) direction and returns toward the initial point, it may be seen that the angle increases by an amount approaching ; and hence as the circuit is completed the logarithm is increased by , the real part of the logarithm returning to its original value. This statement is true, however, only for a path enclosing the origin . If now the point continues to retrace its first path, the logarithm is now given by a different function or, more precisely, by a different "branch" of the same function. That is, if we write

and if on the first circuit is determined (with ) by the branch , then, if is to vary continuously, on the second circuit must be determined by the branch , corresponding to . The point , which must be enclosed by the circuit if transition from one branch to another occurs, is known as a branch point. We may say that the function has infinitely many branches with a single finite branch point at .

The generalized power function , where may be real or complex, is now defined in terms of the logarithm by the equation

If is considered to be real and positive and is real, this definition is clearly in accord with the usual definition if the logarithm of a positive real number is taken to be real. If is a positive integer, this definition must be consistent with (2.14). To see that this is so, we write , where is an integer, and obtain from (2.34)

But since and are integers, we have

and hence there follows, in this case,

This result is in accordance with (2.14), since

More generally, if is a real rational number, we can write

where and are integers having no common factor. Then (2.34) takes the form

or

If we remember that and are given integers, whereas is an arbitrary integer, we can easily see that as takes on any successive integral values, say , the factor will take on corresponding different values, but that if further values of are taken the values so obtained are merely repeated periodically. Hence it follows that for any nonzero value of the function has exactly different values given by

It may be seen also that the function can be considered as having exactly branches, such that if we restrict between and , and if a point traverses a closed contour including the origin (so that changes by ), then a continuous variation of is obtained only through transition from one branch to another. We may verify further that if such a closed contour is traversed exactly times, starting initially with a certain branch, the transition from the branch is back to the initial branch. The point is again a branch point.

As an example, suppose that , and that a point traverses the unit circle in the positive (counterclockwise) direction, starting at the point . If we arbitrarily start out on the branch and note that for there follows , , the initial value of at is given by . As the end of a circuit is approached, the angle approaches , and as the power approaches the value . In order that may vary continuously as the point is passed, and hence drops abruptly to zero and then again increases, we must now determine from the second branch for which , since this branch assumes at the value approached by the first branch when . As the end of the second circuit is approached and again , the power now approaches the value ; and as this point is passed, the transition to the third branch () must occur. Finally, as the end of the third circuit is approached and once more , the power approaches the value , and hence (for continuity) a transition back to the first branch () must take place. The three values of at are

In the general case when may be complex, of the form

where and are real, (2.34) becomes

It is apparent that, in general, the function is infinitely many-valued, with as a branch point. If and is rational, then only a finite number of branches exist; in particular, if and is integral the function is single-valued and is not a branch point.

In place of choosing as the principal value of that value which is in the interval , as in the case of the logarithmic function, one may, for example, equally well adopt the convention in the case of the power function. Then transition from branch to branch will take place along the negative real axis. The choice to be preferred will depend upon the particular application involved.

The generalized exponential function , where may be real or complex, is defined similarly by the equation

If we denote by the principal value of the angle corresponding to , such that , there follows

Although this function is again apparently infinitely many-valued, it is seen that here the ambiguity arises only in the choice of the angle to be associated with the constant , and not upon the specification of the angular position of the point in the plane. That is, here there is no possibility of a continuous transition from an expression corresponding to a given value of to one corresponding to a second value, as the result of motion of a point around a
curve in the complex plane. Thus, in this sense, each choice of can be considered as determining a separate function, rather than a particular branch of a single multivalued function. In any given problem, only one such definition is needed. It is convenient to take , and hence write

We notice in particular that if (and hence ), the definition (2.40) reduces to (2.22). This is the reason for choosing the particular value .

Finally, to conclude the list of elementary functions, we consider the inverse circular and hyperbolic functions. In the case of the inverse sine function, the equation

implies the equation

If we make use of the definition (2.10b), this equation takes the form

or, equivalently,

Equation (2.43) is quadratic in , with the solution

Solving this result for , there follows finally

It is important to notice that if the quantity has two possible values. Then corresponding to each such value the logarithm has infinitely many values. Hence, for any given value of the function has two infinite sets of values. This is, of course, already known to be the case when is real and numerically less than unity so that is real. For example, we have the values or , where in either case may take on arbitrary integral values. In a later section it will be shown that has branch points at .

We may verify that (2.44) gives the known values for by making the calculations

In an entirely analogous way, expressions may be obtained for the other inverse functions. The results may be written in the form

The functions considered in this section are the basic elementary functions. Any linear combination of such functions or any composite function defined in terms of such functions is also known as an elementary function.

The derivative of a function of a complex variable is defined, as in the real case, by the equation

when the indicated limit exists. It is readily verified that the derivative formulas established for elementary functions of a real variable are also valid for the corresponding functions of a complex variable, as defined in this section.

Problems