## 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 [2], may be derived from series [1] by utilizing the relation

and setting *x* = and *x* = , in succession, in series [1]:

Still other series are available which converge even more rapidly than [4].

**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 [3] with §11—21, (II), we note:

*e ^{iθ}* = cos

*θ*+

*i*sin

*θ*, [4]

and, writing — *i* for *i*,

*e ^{−iθ}* = cos

*θ −*

*i*sin

*θ.*[5]

Solving [4] and [5] simultaneously:

Equations [6] and [7] 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 2*θ =* 2 sin *θ* cos θ.

*Solution.* By multiplication, from [6] and [7],

but the right-hand member of (2) is equal (by substitution in [7]) to sin *2θ.* Hence 2 sin *θ* cos *θ* = sin 2*θ.*

EXAMPLE 2. Prove that *e ^{iπ}* + 1 = 0.

*Solution.* In the formula *e ^{iθ}* = cos

*θ + i*sin

*θ*, set

*θ*=

*π*; then

*e ^{iπ}* = cos

*π*+

*i*sin

*π*

*e ^{iπ}* = −1 +

*i*·0

*e ^{iπ}* = −1.

**EXERCISE 11—7**

**1.** Prove, by Euler’s formulas, that sin^{2} *θ* + *cos*^{2} *θ =* 1.

**2.** Prove, by Euler’s formulas, that cos 2*θ* = cos^{2} *θ* – sin^{2} *θ*.

**3.** Prove that *e*^{2iπ} = 1.

**4.** Find the value of *e ^{−iπ};* is this consistent with the value of

*e*found in Illustrative Example 2 above?

^{iπ}**5.** Prove that *e*^{2+iπ} = −*e*^{2}.

**6.** Prove that *e ^{π}*

^{/2}=

**EXERCISE 11—8**

**Review**

**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 *y*^{2} = *x*^{2} *−* *x*^{4} for a singular point at the origin.

**7.** Find the total derivative:

(a) *u* = *2axy* + log *x*; *x* = sin *y.*

(b) *u* = *y*^{2} + *z*^{4} + *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* = *x*^{3} *−* 3*x*^{2} − 24*x*; + 15.

(b) *y* =

(c) *y =* (2 − tan *x) ·* tan *x.*

**9.** Examine the following curves for multiple points:

(a) *y*^{2} = *x*^{3} + 2*x*^{2}.

(b) *y*^{2} = *x* log (1 *+ x*)*.*

**10.** Examine the following curves for cusps:

(a) *x*^{3} = (*y −* *x*)^{2}.

(b) (*x* − *y*)^{2} = (*x* − l)^{5}.

**11.** Prove that:

**12.** Prove that: