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 examples
15. 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.