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217. Functions of a complex variable. In Ch. III we defined the complex variable
and we considered a few simple properties of some classes of expressions involving , such as the polynomial . It is natural to describe such expressions as functions of , and in fact we did describe the quotient , where and are polynomials, as a ‘rational function’. We have however given no general definition of what is meant by a function of .It might seem natural to define a function of in the same way as that in which we defined a function of the real variable , i.e. to say that is a function of if any relation subsists between and in virtue of which a value or values of corresponds to some or all values of . But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if is given, so are and , and conversely: to assign a value of is precisely the same thing as to assign a pair of values of and . Thus a ‘function of ’, according to the definition suggested, is precisely the same thing as a complex function of the two real variables and . For example
are ‘functions of ’. The definition, although perfectly legitimate, is futile because it does not really define a new idea at all. It is therefore more convenient to use the expression ‘function of the complex variable ’ in a more restricted sense, or in other words to pick out, from the general class of complex functions of the two real variables and , a special class to which the expression shall be restricted. But if we were to attempt to explain how this selection is made, and what are the characteristic properties of the special class of functions selected, we should be led far beyond the limits of this book. We shall therefore not attempt to give any general definitions, but shall confine ourselves entirely to special functions defined directly.218. We have already defined polynomials in (§ 39), rational functions of (§ 46), and roots of (§ 47). There is no difficulty in extending to the complex variable the definitions of algebraical functions, explicit and implicit, which we gave (§§ 26–27) in the case of the real variable . In all these cases we shall call the complex number , the argument (§ 44) of the point , the argument of the function under consideration. The question which will occupy us in this chapter is that of defining and determining the principal properties of the logarithmic, exponential, and trigonometrical or circular functions of . These functions are of course so far defined for real values of only, the logarithm indeed for positive values only.
We shall begin with the logarithmic function. It is natural to attempt to define it by means of some extension of the definition
and in order to do this we shall find it necessary to consider briefly some extensions of the notion of an integral.219. Real and complex curvilinear integrals. Let be an arc of a curve defined by the equations
where and are functions of with continuous differential coefficients and ; and suppose that, as varies from to , the point moves along the curve, in the same direction, from to .Then we define the curvilinear integral
(1) |
where and are continuous functions of and , as being equivalent to the ordinary integral obtained by effecting the formal substitutions , , i.e. to
We call the path of integration.Let us suppose now that
so that describes the curve in Argand’s diagram as varies. Further let us suppose that is a polynomial in or rational function of .Then we define
(2) |
as meaning
which is itself defined as meaning220. The definition of . Now let be any complex number. We define , the general logarithm of , by the equation
where is a curve which starts from and ends at and does not pass through the origin. Thus (Fig. 54) the paths (a), (b), (c) are paths such as are contemplated in the definition. The value of is thus defined when the particular path of integration has been chosen. But at present it is not clear how far the value of resulting from the definition depends upon what path is chosen. Suppose for example that is real and positive, say equal to . Then one possible path of integration is the straight line from to , a path which we may suppose to be defined bythe equations , . In this case, and with this particular choice of the path of integration, we have
so that is equal to , the logarithm of according to the definition given in the last chapter. Thus one value at any rate of , when is real and positive, is . But in this case, as in the general case, the path of integration can be chosen in an infinite variety of different ways. There is nothing to show that every value of is equal to ; and in point of fact we shall see that this is not the case. This is why we have adopted the notation , instead of , . is (possibly at any rate) a many valued function, and is only one of its values. And in the general case, so far as we can see at present, three alternatives are equally possible, viz. that(1) we may always get the same value of , by whatever path we go from to ;
(2) we may get a different value corresponding to every different path;
(3) we may get a number of different values each of which corresponds to a whole class of paths:
and the truth or falsehood of any one of these alternatives is in no way implied by our definition.
221. The values of . Let us suppose that the polar coordinates of the point are and , so that
We suppose for the present that , while may have any positive value. Thus may have any value other than zero or a real negative value.The coordinates of any point on the path are functions of , and so also are its polar coordinates . Also
in virtue of the definitions of § 219. But , , and
so that
where denotes the difference between the values of at the points corresponding to and , and has a similar meaning.It is clear that
but the value of requires a little more consideration. Let us suppose first that the path of integration is the straight line from to . The initial value of is the amplitude of , or ratherone of the amplitudes of , viz. , where is any integer. Let us suppose that initially . It is evident from the figure that increases from to as moves along the line. Thus
and, when the path of integration is a straight line, .We shall call this particular value of the principal value. When is real and positive, and , so that the principal value of is the ordinary logarithm . Hence it will be convenient in general to denote the principal value of by . Thus
and the principal value is characterised by the fact that its imaginary part lies between and .Next let us consider any path (such as those shown in Fig. 56) such that the area or areas included
between the path and the straight line from to does not include the origin. It is easy to see that is still equal to . Along the curve shown in the figure by a continuous line, for example, , initially equal to , first decreases to the value
and then increases again, being equal to at , and finally to . The dotted curve shows a similar but slightly more complicated case in which the straight line and the curve bound two areas, neither of which includes the origin. Thus if the path of integration is such that the closed curve formed by it and the line from to does not include the origin, thenOn the other hand it is easy to construct paths of integration such that is not equal to . Consider, for example, the curve indicated by a continuous line in Fig. 57. If is initially equal to , it will have increased
by when we get to and by when we get to ; and its final value will be , so that and
In this case the path of integration winds twice round the origin in the positive sense. If we had taken a path winding times round the origin we should have found, in a precisely similar way, that and
Here is positive. By making the path wind round the origin in the opposite direction (as shown in the dotted path in Fig. 57), we obtain a similar series of values in which is negative. Since , and the different angles are the different values of , we conclude that every value of is a value of ; and it is clear from the preceding discussion that every value of must be of this form.We may summarise our conclusions as follows: the general value of is where is any positive or negative integer. The value of is determined by the path of integration chosen. If this path is a straight line then and
In what precedes we have used to denote the argument of the function , and or to denote the coordinates of ; and , , to denote an arbitrary point on the path of integration and its coordinates. There is however no reason now why we should not revert to the natural notation in which is used as the argument of the function , and we shall do this in the following examples.
Examples XCIII. 1. We supposed above that , and so excluded the case in which is real and negative. In this case the straight line from to passes through , and is therefore not admissible as a path of integration. Both and are values of , and is equal to one or other of them: also . The values of are still the values of , viz.
where is an integer. The values and correspond to paths from to lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of , as convenience dictates. We shall choose the value i corresponding to the first path.2. The real and imaginary parts of any value of are both continuous functions of and , except for , .
3. The functional equation satisfied by . The function satisfies the equation
(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
and applying the formula of p. 1364. It is however not true that(2) |
in all circumstances. If, e.g.,
then , and , which is one of the values of , but not the principal value. In fact .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 , where 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 is completely true. It is also true that , except when is real and negative.
6. The equation
is true if lies outside the region bounded by the line joining the points , , and lines through these points parallel to and extending to infinity in the negative direction.7. The equation
is true if lies outside the triangle formed by the three points , , .8. Draw the graph of the function of the real variable . [The graph consists of the positive halves of the lines and the negative halves of the lines .]
9. The function of the real variable , defined by
is equal to when is positive and to when is negative.10. The function defined by
is equal to when , to when , and to when .11. For what values of is (i) (ii) any value of (a) real or (b) purely imaginary?
12. If then , where
and is the least positive angle determined by the equations Plot roughly the doubly infinite set of values of , indicating which of them are values of and which of .222. The exponential function. In Ch. IX we defined a function of the real variable as the inverse of the function . It is naturally suggested that we should define a function of the complex variable which is the inverse of the function .
DEFINITION. If any value of is equal to , we call the exponential of and write
Thus if . It is certain that to any given value of correspond infinitely many different values of . It would not be unnatural to suppose that, conversely, to any given value of correspond infinitely many values of , or in other words that is an infinitely many-valued function of . This is however not the case, as is proved by the following theorem.
THEOREM. The exponential function is a one-valued function of .
For suppose that
are both values of . Then and so where and are integers. This involves Thus , and and differ by a multiple of . Hence .COROLLARY. If is real then , the real exponential function of defined in Ch. IX.
For if then , i.e. one of the values of is . Hence .
Then where is an integer. Hence , , or and accordinglyIf then , as we have already inferred in § 222. It is clear that both the real and the imaginary parts of are continuous functions of and for all values of and .
224. The functional equation satisfied by . Let , . Then
The exponential function therefore satisfies the functional relation , an equation which we have proved already (§ 205) to be true for real values of and .
225. The general power . It might seem natural, as when is real, to adopt the same notation when is complex and to drop the notation altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol : we shall find then that represents a function with infinitely many values of which is only one.
We have already defined the meaning of the symbol in a considerable variety of cases. It is defined in elementary Algebra in the case in which is real and positive and rational, or real and negative and a rational fraction whose denominator is odd. According to the definitions there given has at most two values. In Ch. III we extended our definitions to cover the case in which is any real or complex number and any rational number ; and in Ch. IX we gave a new definition, expressed by the equation
which applies whenever is real and real and positive.Thus we have, in one way or another, attached a meaning to such expressions as
but we have as yet given no definitions which enable us to attach any meaning to such expressions as We shall now give a general definition of which applies to all values of and , real or complex, with the one limitation that must not be equal to zero.DEFINITION. The function is defined by the equation where is any value of the logarithm of .
We must first satisfy ourselves that this definition is consistent with the previous definitions and includes them all as particular cases.
(1) If is positive and real, then one value of , viz. , is real: and , which agrees with the definition adopted in Ch. IX. The definition of Ch. IX is, as we saw then, consistent with the definition given in elementary Algebra; and so our new definition is so too.
(2) If , then
where may have any integral value. It is easy to see that if assumes all possible integral values then this expression assumes and only different values, which are precisely the values of found in § 48. Hence our new definition is also consistent with that of Ch. III.
226. The general value of . Let
where , so that, in the notation of § 225, or .Then
where and Thus the general value of isIn general is an infinitely many-valued function. For
has a different value for every value of , unless . If on the other hand , then the moduli of all the different values of are the same. But any two values differ unless their amplitudes are the same or differ by a multiple of . This requires that and , where and are different integers, shall differ, if at all, by a multiple of . But if then is rational. We conclude that is infinitely many-valued unless is real and rational. On the other hand we have already seen that, when is real and rational, has but a finite number of values.The principal value of is obtained by giving its principal value, i.e. by supposing in the general formula. Thus the principal value of is
Two particular cases are of especial interest. If is real and positive and real, then , , , , and the principal value of is , which is the value defined in the last chapter. If and is real, then , , , and the principal value of is . This is a further generalisation of De Moivre’s Theorem (§§45, 49).
Examples XCIV. 1. Find all the values of . [By definition
But where is any integer. Hence All the values of are therefore real and positive.]2. Find all the values of , , .
3. The values of , when plotted in the Argand diagram, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of .
[If we have
and all the points lie on the spiral .]4. The function . If we write for in the general formula, so that , , we obtain
The principal value of is , which is equal to (§223). In particular, if is real, so that , we obtain as the general and as the principal value, denoting here the positive value of the exponential defined in Ch. IX.5. Show that , where and are any integers, and that in general has a double infinity of values.
6. The equation is completely true (Ex. XCIII. 3): it is also true of the principal values.
7. The equation is completely true but not always true of the principal values.
8. The equation is not completely true, but is true of the principal values. [Every value of the right-hand side is a value of the left-hand side, but the general value of , viz.
is not as a rule a value of unless .]9. What are the corresponding results as regards the equations
10. For what values of is (a) any value (b) the principal value of (i) real (ii) purely imaginary (iii) of unit modulus?
11. The necessary and sufficient conditions that all the values of should be real are that and , where denotes any value of the amplitude, should both be integral. What are the corresponding conditions that all the values should be of unit modulus?
12. The general value of , where , is
13. Explain the fallacy in the following argument: since , where and are any integers, therefore, raising each side to the power we obtain .
14. In what circumstances are any of the values of , where is real, themselves real? [If then
the first factor being real. The principal value, for which , is always real.If is a rational fraction , or is irrational, then there is no other real value. But if is of the form , then there is one other real value, viz. , given by .
If then
The only case in which any value is real is that in which , when gives the real value The cases of reality are illustrated by the examples15. Logarithms to any base. We may define in two different ways. We may say (i) that if the principal value of is equal to ; or we may say (ii) that if any value of is equal to .
Thus if then , according to the first definition, if the principal value of is equal to , or if ; and so is identical with . But, according to the second definition, if
or , any values of the logarithms being taken. Thus so that is a doubly infinitely many-valued function of . And generally, according to this definition, .16. , , where and are any integers.
227. The exponential values of the sine and cosine. From the formula
we can deduce a number of extremely important subsidiary formulae. Taking , we obtain ; and, changing the sign of , . HenceWe can of course deduce expressions for any of the trigonometrical ratios of in terms of .
228. Definition of and for all values of . We saw in the last section that, when is real,
The left-hand sides of these equations are defined, by the ordinary geometrical definitions adopted in elementary Trigonometry, only for real values of . The right-hand sides have, on the other hand, been defined for all values of , real or complex. We are therefore naturally led to adopt the formulae (1) as the definitions of and for all values of . These definitions agree, in virtue of the results of § 227, with the elementary definitions for real values of .
Having defined and , we define the other trigonometrical ratios by the equations
(2) |
It is evident that and are even functions of , and , , , and odd functions. Also, if , we have
We can moreover express the trigonometrical functions of in terms of those of and by precisely the same formulae as those which hold in elementary trigonometry. For if , , we have
and similarly we can prove that
(5) |
In particular
(6) |
All the ordinary formulae of elementary Trigonometry are algebraical corollaries of the equations (2)–(6); and so all such relations hold also for the generalised trigonometrical functions defined in this section.
229. The generalised hyperbolic functions. In Ex. LXXXVII. 19, we defined and , for real values of , by the equations
(1) |
We can now extend this definition to complex values of the variable; i.e. we can agree that the equations (1) are to define and for all values of real or complex. The reader will easily verify the following relations:
We have seen that any elementary trigonometrical formula, such as the formula , remains true when is allowed to assume complex values. It remains true therefore if we write for , for and for ; or, in other words, if we write for , for , and for . Hence
The same process of transformation may be applied to any trigonometrical identity. It is of course this fact which explains the correspondence noted in Ex. LXXXVII. 21 between the formulae for the hyperbolic and those for the ordinary trigonometrical functions.230. Formulae for , , etc. It follows from the addition formulae that
These formulae are true for all values of and . The interesting case is that in which and are real. They then give expressions for the real and imaginary parts of the cosine and sine of a complex number.
Examples XCV. 1. Determine the values of for which and are (i) real (ii) purely imaginary. [For example is real when or when is any multiple of .]
2.
[Use (e.g.) the equation .]
3. , .
[For example
which leads at once to the result given.]4.
5. If then , and if then .
6. If , then
7. Prove that , where
and is any angle such that Find a similar formula for .8. Solution of the equation , where is real. Putting , and equating real and imaginary parts, we obtain
Hence either or is a multiple of . If (i) then , which is impossible unless . This hypothesis leads to the solution where lies between and . If (ii) then , so that either and is even, or and is odd. If then , and we are led back to our first case. If then , and we are led to the solutionsFor example, the general solution of is .
9. Solve , where is real.
10. Solution of , where . We may suppose , since the results when may be deduced by merely changing the sign of . In this case
(1) |
and
If we put we find that
or , where Suppose . Then and . Also and since we must take The general solutions of these equations are(2) |
where , , and lies between and .
The values of and thus found above include, however, the solutions of the equations
(3) |
as well as those of the equations (1), since we have only used the second of the latter equations after squaring it. To distinguish the two sets of solutions we observe that the sign of is the same as the ambiguous sign in the first of the equations (2), and the sign of is the same as the ambiguous sign in the second. Since , these two signs must be different. Hence the general solution required is
11. Work out the cases in which and in the same way.
12. If then and . Verify that the results thus obtained agree with those of Ex. 8.
13. Showthatif and are positive then the general solution of is
where lies between and . Obtain the solution in the other possible cases.14. Solve , where is real. [All the roots are real.]
15. Show that the general solution of , where , is
where is the numerically least angle such that16. If , where is real, and is also real, then the modulus of is
17. Prove that
18. Prove that tends to if moves away towards infinity along any straight line through the origin making an angle less than with , and to if moves away along a similar line making an angle greater than with .
19. Prove that and tend to if moves away towards infinity along any straight line through the origin other than either half of the real axis.
20. Prove that tends to or to if moves away to infinity along the straight line of Ex. 19, to if the line lies above the real axis and to if it lies below.
231. The connection between the logarithmic and the inverse trigonometrical functions. We found in Ch. VI that the integral of a rational or algebraical function , where , , … are constants, often assumes different forms according to the values of , , …; sometimes it can be expressed by means of logarithms, and sometimes by means of inverse trigonometrical functions. Thus, for example,
(1) |
if , but
(2) |
if . These facts suggest the existence of some functional connection between the logarithmic and the inverse circular functions. That there is such a connection may also be inferred from the facts that we have expressed the circular functions of in terms of , and that the logarithm is the inverse of the exponential function.
Let us consider more particularly the equation
which holds when is real and is positive. If we could write instead of in this equation, we should be led to the formula(3) |
where is a constant, and the question is suggested whether, now that we have defined the logarithm of a complex number, this equation will not be found to be actually true.
Now (§221)
where is an integer and is the numerically least angle such that and . Thus where is an integer, and this does in fact differ by a constant from any value of .The standard formula connecting the logarithmic and inverse circular functions is
(4) |
where is real. It is most easily verified by putting , when the right-hand side reduces to
where is any integer, so that the equation (4) is ‘completely’ true (Ex. XCIII. 3). The reader should also verify the formulae(5) |
where : each of these formulae also is ‘completely’ true.
Example. Solving the equation
where , with respect to , we obtain . Thus: which is equivalent to the first of the equations (5). Obtain the remaining equations (4) and (5) by similar reasoning.232. The power series for .109 We saw in § 212 that when is real
(1) |
Moreover we saw in § 191 that the series on the right-hand side remains convergent (indeed absolutely convergent) when is complex. It is naturally suggested that the equation (1) also remains true, and we shall now prove that this is the case.
Let the sum of the series (1) be denoted by . The series being absolutely convergent, it follows by direct multiplication (as in Ex. LXXXI. 7) that satisfies the functional equation
(2) |
Now let , where is real, and . Then
and soBut
and so, if , Hence as , and so(3) |
Now
where is an even and an odd function of , and soand therefore
where is a function of such that . Since has a differential coefficient, its real and imaginary parts and have differential coefficients, and are a fortiori continuous functions of . Hence is a continuous function of . Suppose that changes to when changes to . Then tends to zero with , and Of the two quotients on the right-hand side the first tends to a limit when , since has a differential coefficient with respect to , and the second tends to the limit . Hence tends to a limit, so that has a differential coefficient with respect to .Further
But we have seen already that Hence where is a constant, andBut when , so that is a multiple of , and . Thus for all real values of . And, if also is real, we have
or for all values of .233. The power series for and . From the result of the last section and the equations (1) of § 228 it follows at once that
for all values of . These results were proved for real values of in Ex. LVI. 1.Examples XCVI. 1. Calculate and to two places of decimals by means of the power series for and .
2. Prove that and .
3. Prove that if then and .
4. Since we have
Prove by multiplying the two series on the right-hand side (§195) and equating coefficients (§194) that Verify the result by means of the binomial theorem. Derive similar identities from the equations5. Show that
6. Expand in powers of . [We have
and similarly
Hence7. Expand , , and in powers of .
8. Expand and in powers of . [Use the formulae
It is clear that the same method may be used to expand and , where is any integer.]9. Sum the series
[Here
and similarly
Hence10. Sum
11. Sum
and the corresponding series involving sines.12. Show that
13. Show that the expansions of and in powers of (Ex. LVI. 1) are valid for all values of and , real or complex.
234. The logarithmic series. We found in § 213 that
(1) |
when is real and numerically less than unity. The series on the right-hand side is convergent, indeed absolutely convergent, when has any complex value whose modulus is less than unity. It is naturally suggested that the equation (1) remains true for such complex values of . That this is true may be proved by a modification of the argument of § 213. We shall in fact prove rather more than this, viz. that (1) is true for all values of such that , with the exception of the value .
It will be remembered that is the principal value of , and that
where is the straight line joining the points and in the plane of the complex variable . We may suppose that is not real, as the formula (1) has been proved already for real values of .If we put
so that , and then will describe as increases from to . Andwhere
(3) |
It follows from (1) of § 164 that
(4) |
Now or is never less than , the perpendicular from on to the line .110 Hence
and so as . It follows from (2) that(5) |
We have of course shown in the course of our proof that the series is convergent: this however has been proved already (Ex. LXXX. 4). The series is in fact absolutely convergent when and conditionally convergent when .
Changing into we obtain
(6) |
That value of the inverse tangent must be taken which lies between and . For, since is the vector represented by the line from to , the principal value of always lies between these limits when lies within the circle .111
Since , we obtain, on equating the real and imaginary parts in equation (5) of § 234,
These equations hold when , and for all values of , except that, when , must not be equal to an odd multiple of . It is easy to see that they also hold when , except that, when , must not be equal to an even multiple of .
A particularly interesting case is that in which . In this case we have
if , and so
The sums of the series, for other values of , are easily found from the consideration that they are periodic functions of with the period . Thus the sum of the cosine series is for all values of save odd multiples of (for which values the series is divergent), while the sum of the sine series is if , and zero if is an odd multiple of . The graph of the function represented by the sine series is shown in Fig. 58. The function is discontinuous for .
If we write and for in (5), and subtract, we obtain
If is real and numerically less than unity, we are led, by the results of §231, to the formula already proved in a different manner in §214.Examples XCVII. 1. Prove that, in any triangle in which ,
[Use the formula .]
2. Prove that if and then
the inverse tangent lying between and . Determine the sum of the series for all other values of .3. Prove, by considering the expansions of and in powers of , that if then
the inverse tangents lying between and .
4. Prove that
the inverse cotangent lying between and ; and find similar expressions for the sums of the series
236. Some applications of the logarithmic series. The exponential limit. Let be any complex number, and a real number small enough to ensure that . Then
and so whereso that as . It follows that
(1) |
If in particular we suppose , where is a positive integer, we obtain
and so(2) |
This is a generalisation of the result proved in § 208 for real values of .
From (1) we can deduce some other results which we shall require in the next section. If and are real, and is sufficiently small, we have
which tends to the limit as . Hence(3) |
We shall also require a formula for the differentiation of , where is any number real or complex, with respect to . We observe first that, if is a complex function of , whose real and imaginary parts and possess derivatives, then
so that the rule for differentiating is the same as when is real. This being so we have
Here both and have their principal values.
237. The general form of the Binomial Theorem. We have proved already (§ 215) that the sum of the series
is , for all real values of and all real values of between and . If is the coefficient of then whether is real or complex. Hence (Ex. LXXX. 3) the series is always convergent if the modulus of is less than unity, and we shall now prove that its sum is still , i.e. the principal value of .It follows from § 236 that if is real then
and having any real or complex values and each side having its principal value. Hence, if , we have This formula still holds if , so thatNow, in virtue of the remark made at the end of § 164, we have
where But if then and thereforewhere ; so that (cf. § 163)
say. But and so (Ex. XXVII. 6) , and therefore , as . Hence we arrive at the following theorem.THEOREM. The sum of the binomial series is , where the logarithm has its principal value, for all values of , real or complex, and all values of such that .
A more complete discussion of the binomial series, taking account of the more difficult case in which , will be found on pp. 225 et seq. of Bromwich’s Infinite Series.
Examples XCVIII. 1. Suppose real. Then since
we obtainall the inverse tangents lying between and . In particular, if we suppose , , and equate the real and imaginary parts, we obtain
2. Verify the formulae of Ex. 1 when , , . [Of course when is a positive integer the series is finite.]
3. Prove that if then
[Take in the last two formulae of Ex. 1.]
4. Prove that if then
for all real values of . [These results follow at once from the equations
5. We proved (Ex. LXXXI. 6), by direct multiplication of series, that , where , satisfies the functional equation
Deduce, by an argument similar to that of §216, and without assuming the general result of p. 1454, that if is real and rational then6. If and are real, and , then
1. Show that the real part of is
where is any integer.2. If , where , , are real and , then
where is any odd or any even integer, according as is positive or negative, and is an angle whose cosine and sine are and .3. Prove that if is real and then
where is any even or any odd integer, according as is positive or negative.4. Show that if is real then
Deduce the results of Ex. LXXXVII. 3.
5. Show that if then , and deduce the results of Ex. LXXXVII. 5.
6. Show that if is the equation of an ellipse, and denotes the terms of highest degree in the equation of any other algebraic curve, then the sum of the eccentric angles of the points of intersection of the ellipse and the curve differs by a multiple of from
[The eccentric angles are given by or by
where ; and is equal to one of the values of , where is the product of the roots of this equation.]7. Determine the number and approximate positions of the roots of the equation , where is real.
[We know already (Ex. XVII. 4) that the equation has infinitely many real roots. Now let , and equate real and imaginary parts. We obtain
so that, unless or is zero, we have This is impossible, the left-hand side being numerically less, and the right-hand side numerically greater than unity. Thus or . If we come back to the real roots of the equation. If then . It is easy to see that this equation has no real root other than zero if or , and two such roots if . Thus there are two purely imaginary roots if ; otherwise all the roots are real.]8. The equation , where and are real and is not equal to zero, has no complex roots if . If then the real parts of all the complex roots are numerically greater than .
9. The equation , where is real, has no complex roots, but has two purely imaginary roots if .
10. The equation , where and are real, has an infinity of real and of purely imaginary roots, but no complex roots.
11. Show that if is real then
where there are or terms inside the large brackets. Find a similar series for .12. If as , then .
13. If is a complex function of the real variable , then
[Use the formulae
14. Transformations. In Ch. III (Exs. XXI. 21 et seq., and Misc. Exs. 22 et seq.) we considered some simple examples of the geometrical relations between figures in the planes of two variables , connected by a relation . We shall now consider some cases in which the relation involves logarithmic, exponential, or circular functions.
Suppose firstly that
where is positive. To one value of corresponds one of , but to one of infinitely many of . If , , , are the coordinates of and , , , those of , we have the relationswhere is any integer. If we suppose that , and that has its principal value , then , and is confined to a strip of its plane parallel to the axis and extending to a distance from it on each side, one point of this strip corresponding to one of the whole -plane, and conversely. By taking a value of other than the principal value we obtain a similar relation between the -plane and another strip of breadth in the -plane.
To the lines in the -plane for which and are constant correspond the circles and radii vectores in the -plane for which and are constant. To one of the latter lines corresponds the whole of a parallel to , but to a circle for which is constant corresponds only a part, of length , of a parallel to . To make describe the whole of the latter line we must make move continually round and round the circle.
15. Show that to a straight line in the -plane corresponds an equiangular spiral in the -plane.
16. Discuss similarly the transformation , showing in particular that the whole -plane corresponds to any one of an infinite number of strips in the -plane, each parallel to the axis and of breadth . Show also that to the line corresponds the ellipse
and that for different values of these ellipses form a confocal system; and that the lines correspond to the associated system of confocal hyperbolas. Trace the variation of as describes the whole of a line or . How does vary as describes the degenerate ellipse and hyperbola formed by the segment between the foci of the confocal system and the remaining segments of the axis of ?17. Verify that the results of Ex. 16 are in agreement with those of Ex. 14 and those of Ch. III, Misc. Ex. 25. [The transformation may be regarded as compounded from the transformations
18. Discuss similarly the transformation , showing that to the lines correspond the coaxal circles
and to the lines the orthogonal system of coaxal circles.19. The Stereographic and Mercator’s Projections. The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are , , ) on to the tangent plane at the north pole. The coordinates of a point on the sphere are , , , and Cartesian axes , are taken on the tangent plane, parallel to the axes of and . Show that the coordinates of the projection of the point are
and that , where is the longitude (measured from the plane ) and the north polar distance of the point on the sphere.This projection gives a map of the sphere on the tangent plane, generally known as the Stereographic Projection. If now we introduce a new complex variable
so that , , we obtain another map in the plane of , usually called Mercator’s Projection. In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of and respectively.20. Discuss the transformation given by the equation
showing that the straight lines for which and are constant correspond to two orthogonal systems of coaxal circles in the -plane.21. Discuss the transformation
showing that the straight lines for which and are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points and .[We have
and it will be found that
22. The transformation . If , where the imaginary power has its principal value, we have
so that , , where is an integer. As all values of give the same point , we shall suppose that , so that(1) |
The whole plane of is covered when varies through all positive values and from to : then has the range to and ranges through all real values. Thus the -plane corresponds to the ring bounded by the circles , ; but this ring is covered infinitely often. If however is allowed to vary only between and , so that the ring is covered only once, then can vary only from to , so that the variation of is restricted to a ring similar in all respects to that within which varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which (or ) must not cross, as its amplitude must not transgress the limits and .
We thus obtain a correspondence between two rings, given by the pair of equations
where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other.23. Trace the variation of when , starting at the point , moves round the larger circle in the positive direction to the point , along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position.
24. Suppose each plane to be divided up into an infinite series of rings by circles of radii
Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations , .25. If , any value of the power being taken, and moves along an equiangular spiral whose pole is the origin in its plane, then moves along an equiangular spiral whose pole is the origin in its plane.
26. How does , where is real, behave as approaches the origin along the real axis? [ moves round and round a circle whose centre is the origin (the unit circle if has its principal value), and the real and imaginary parts of both oscillate finitely.]
27. Discuss the same question for , where and are any real numbers.
28. Show that the region of convergence of a series of the type , where is real, is an angle, i.e. a region bounded by inequalities of the type [The angle may reduce to a line, or cover the whole plane.]
29. Level Curves. If is a function of the complex variable , we call the curves for which is constant the level curves of . Sketch the forms of the level curves of
30. Sketch the forms of the level curves of , . [Some of the level curves of the latter function are drawn in Fig. 59, the curves marked I–VII corresponding to the values
of . The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.]31. Sketch the forms of the level curves of (i) , (ii) . [See Fig. 60, which represents the level curves of . The curves marked I–VIII correspond to , , , , , , , .]
32. Sketch the forms of the level curves of , where is a real constant. [Fig. 61 shows the level curves of , the curves I–VII corresponding to the values of given by , , , , , , .]
33. The level curves of , where is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether or . In Fig. 62 we have taken , and the curves I–VIII correspond to , , , , , , , . In Fig. 63 we have taken , and the curves I–VII correspond to , , , , , , . If then the curves are the same as those of Fig. 60, except that the origin and scale are different.]
34. Prove that if then
and determine the sums of the series for all other values of for which they are convergent. [Use the equation
where . When is increased by the sum of each series simply changes its sign. It follows that the first formula holds for all values of save multiples of (for which the series diverges), while the sum of the second series is if , if , and if is a multiple of .]35. Prove that if then
and determine the sums of the series for all other values of for which they are convergent.
36. Prove that
unless or is a multiple of .37. Prove that if neither nor is real then
each logarithm having its principal value. Verify the result when , , where is positive. Discuss also the cases in which or or both are real and negative.38. Prove that if and are real, and , then
What is the value of the integral when ?39. Prove that, if the roots of have their imaginary parts of opposite signs, then
the sign of being so chosen that the real part of is positive.
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