The Calculus Primer (2011)
Part XI. Expansion of Functions
Chapter 43. THE VALUE OF π; EULER’S FORMULA
11—22. Calculating Tables of Functions. Power series such as the expansion of sin x and cos x in §11—21, (II), are frequently used when computing tables of values of functions such as logarithms, trigonometric functions, exponential functions, etc. The two series in question are convergent for all finite values of x, where x is expressed in radian measure.
To compute the numerical value of π, any one of several series may be employed. For example, Gregory’s series,
where θ is an angle such that tan θ 1, can be so used, since it holds for values of θ between + and − inclusive. If we set tan θ = x, where − 1 < x < +1, we get:
setting x = 1, we have:
Unfortunately, since this series converges very slowly, it is not practical, for a great many terms would have to be added to obtain the value of π to any great degree of accuracy.
Another series, known as Euler’s series, which converges somewhat more rapidly than series , may be derived from series  by utilizing the relation
and setting x = and x = , in succession, in series :
Still other series are available which converge even more rapidly than .
11—23. Euler’s Analytic Formulas for the Sine and Cosine. Consider the exponential series
and set x = iθ, where i = :
Grouping the real and imaginary terms, we have
Comparing  with §11—21, (II), we note:
eiθ = cos θ + i sin θ, 
and, writing — i for i,
e−iθ = cos θ − i sin θ. 
Solving  and  simultaneously:
Equations  and  are Euler’s analytic definitions for the sine and cosine functions; they have many uses in higher mathematics.
EXAMPLE 1. Using Euler’s formulas, prove that sin 2θ = 2 sin θ cos θ.
Solution. By multiplication, from  and ,
but the right-hand member of (2) is equal (by substitution in ) to sin 2θ. Hence 2 sin θ cos θ = sin 2θ.
EXAMPLE 2. Prove that eiπ + 1 = 0.
Solution. In the formula eiθ = cos θ + i sin θ, set θ = π; then
eiπ = cos π + i sin π
eiπ = −1 + i·0
eiπ = −1.
1. Prove, by Euler’s formulas, that sin2 θ + cos2 θ = 1.
2. Prove, by Euler’s formulas, that cos 2θ = cos2 θ – sin2 θ.
3. Prove that e2iπ = 1.
4. Find the value of e−iπ; is this consistent with the value of eiπ found in Illustrative Example 2 above?
5. Prove that e2+iπ = −e2.
6. Prove that eπ/2 =
1. Write the first five terms of each of the following infinite series with the given general term:
2. Determine whether the following series are convergent or divergent:
3. Show that each of the following series is convergent, and determine in each case the interval of convergence:
4. Find the limiting value of:
(a) sin x log cot x.
(b) (1 + sin θ)θ.
5. Find the altitude x of the rectangle of maximum area that can be inscribed in an acute-angled triangle of altitude h.
6. Test the equation y2 = x2 − x4 for a singular point at the origin.
7. Find the total derivative:
(a) u = 2axy + log x; x = sin y.
(b) u = y2 + z4 + zy; y = sin x, z = cos x.
8. Find the values of x for which the following functions have a maximum or minimum value, and indicate which:
(a) y = x3 − 3x2 − 24x; + 15.
(b) y =
(c) y = (2 − tan x) · tan x.
9. Examine the following curves for multiple points:
(a) y2 = x3 + 2x2.
(b) y2 = x log (1 + x).
10. Examine the following curves for cusps:
(a) x3 = (y − x)2.
(b) (x − y)2 = (x − l)5.
11. Prove that:
12. Prove that: