The Constant Function c

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Lacking dependence on even a single variable, the constant function is the simplest, and an almost trivial, function.

1.1 Notation

Constants are also known as invariants and are represented by a variety of symbols, mostly letters drawn from early members of the Latin and Greek alphabets. In this chapter, we mostly employ c to represent an arbitrary constant.

1.2 Behavior

Figure 1-1 is a graphical representation of the constant function f(x) = c, a horizontal line extending to x = ±∞, reflecting the fact that f takes the same value for all x.

Figure 1-1

Figure 1-1

1.3 Definitions

The constant function is defined for all values of its argument x and has the same value, c, irrespective of x.

1.4 Special Cases

When c is zero, the constant function is sometimes termed the zero function. Likewise, the function f(x) = c = 1 is sometimes known as the unit function or unity function.

1.5 Intrarelationships

Being relations between function values at different values of the argument, intrarelationships are of no consequence for the constant function.

1.6 Expansions

A constant may be represented as a finite sum by utilizing the formulas for an arithmetic series:

$$\begin{array}{c} c = \upalpha + (\upalpha + \delta ) + (\upalpha + 2\delta ) + \cdots + (\upalpha + J\delta ) = \sum\limits_{j = 0}^J {(\upalpha + j\delta )} \\ \upalpha = \frac{c}{{J + 1}} - \frac{{J\delta }}{2}\quad \quad {\rm{or}}\quad \quad c = (J + 1)\left( {\upalpha + \frac{{J\delta }}{2}} \right) \\ \end{array}$$

(1:6:1)

a geometric series:

$$\begin{array}{c} c = \upalpha + \upalpha \upbeta + \upalpha \upbeta ^2 + \cdots + \upalpha \upbeta ^J = \sum\limits_{j = 0}^J {\upalpha \upbeta ^j } \\ \upalpha = c\frac{{\upbeta - 1}}{{\upbeta ^{J + 1} - 1}}\quad \quad {\rm{or}}\quad \quad c = \upalpha \frac{{\upbeta ^{J + 1} - 1}}{{\upbeta - 1}} \\ \end{array}$$

(1:6:2)

or an arithmetic-geometric series:

$$\begin{array}{c} c = \upalpha + \upbeta (\upalpha + \delta ) + \upbeta ^2 (\upalpha + 2\delta ) + \cdots + \upbeta ^J (\upalpha + J\delta ) = \sum\limits_{j = 0}^J {\upbeta ^j (\upalpha + j\delta )} \\ \upalpha = \frac{{c(\upbeta - 1) - J\delta \upbeta ^{J + 1} + \delta \upbeta (\upbeta ^J - 1)/(\upbeta - 1)}}{{\upbeta ^{J - 1} - 1}}\quad \quad {\rm{or}}\quad \quad c = \left( {\upbeta ^{J + 1} - 1} \right)\left[ {\upbeta (\upalpha + \delta ) - \upalpha } \right] + J(\upbeta - 1)\upbeta ^{J + 1} \delta \\ \end{array}$$

(1:6:3)

In these formulas β and δ are arbitrary and J may be any positive integer.

Any constant greater than ½ may be expanded as the infinite geometric sum

$$c = 1 + \left( {\frac{{c - 1}}{c}} \right) + \left( {\frac{{c - 1}}{c}} \right)^2 + \left( {\frac{{c - 1}}{c}} \right)^3 + \cdots \; = \sum\limits_{j = 1}^\infty {\left( {\frac{{c - 1}}{c}} \right)^j \quad \quad \quad c > {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} $$

(1:6:4)

or as the infinite product

$$c = \left[ {1 + \left( {\frac{{c - 1}}{c}} \right)} \right]\left[ {1 + \left( {\frac{{c - 1}}{c}} \right)^2 } \right]\left[ {1 + \left( {\frac{{c - 1}}{c}} \right)^4 } \right] \cdots \; = \prod\limits_{j = 0}^\infty {\left[ {1 + \left( {\frac{{c - 1}}{c}} \right)^{2^j } } \right]} \quad \quad \quad c > {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$$

(1:6:5)

α	β	constraint
c	1 − c	−1 ≤ c < 1
1	$\frac{{1 - c^2 }}{c}$	0 < c 2 < 1
c 2 + c	1	c ≥ − ½

A constant is expansible as the infinite continued fraction

$$c = \frac{\upalpha }{{\upbeta + }}\frac{\upalpha }{{\upbeta + }}\frac{\upalpha }{{\upbeta + }} \cdots $$

(1:6:6)

in the variety of ways indicated in the table, which lists three alternative assignments of the terms α and β, any one of which validates expansion 1:6:6.

1.7 Particular Values

Certain constants occur frequently in the theory of functions. Four of these – Archimedes’s constant, Catalan’s constant, the base of natural logarithms and Euler’s constant – are important irrational numbers. There are many formulations of these four constants other than the ones we present here; see Gradshteyn and Ryzhik [Chapter 0] for some of these.

Archimedes (Archimedes of Syracuse, Greek philosopher, 287−212 BC) himself was content merely to bracket his constant by (223/71) < π < (22/7). It was the sixteenth-century Frenchman François Viète (“Vieta”) who discovered the first formula

$$\pi = \frac{2}{{\sqrt {{\textstyle{1 \over 2}}} \times \sqrt {{\textstyle{1 \over 2}} + {\textstyle{1 \over 2}}\sqrt {{\textstyle{1 \over 2}}} } \times \sqrt {{\textstyle{1 \over 2}} + {\textstyle{1 \over 2}}\sqrt {{\textstyle{1 \over 2}} + {\textstyle{1 \over 2}}\sqrt {{\textstyle{1 \over 2}}} } } \times \cdots }} = 3.1415\;92653\;58979$$

(1:7:1)

for Archimedes’s constant, also known simply as pi. It may also be defined by the infinite sum

$$\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots = 4\sum\limits_{j = 0}^\infty {\frac{{( - 1)^j }}{{2j + 1}} = 3.1415\;92653\;58979} $$

(1:7:2)

discovered by Gregory (James Gregory, Scottish mathematician, 1638 −1675), as the infinite product

$$\pi = 2 \times \frac{4}{3} \times \frac{{16}}{{15}} \times \frac{{36}}{{35}} \times \frac{{64}}{{63}} \times \cdots = 2\prod\limits_{j = 1}^\infty {\frac{{j^2 }}{{j^2 - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}}}} = 3.1415\;92653\;58979$$

(1:7:3)

and in numerous other ways. The definition of Catalan’s constant (Eugène Charles Catalan, Belgian mathematician 1814 −1894) is similar to 1:7:2

$$G = 1 - \frac{1}{9} + \frac{1}{{25}} - \frac{1}{{49}} + \cdots = \sum\limits_{j = 0}^\infty {\frac{{( - 1)^j }}{{\left( {2j + 1} \right)^2 }} = 0.91596\;55941\;77219} $$

(1:7:4)

The base of natural logarithms may be defined as a sum of all reciprocal factorial functions [Chapter 2]

$$e = 1 + \frac{1}{1} + \frac{1}{{1 \times 2}} + \frac{1}{{1 \times 2 \times 3}} + \cdots = \sum\limits_{j = 0}^\infty {\frac{1}{{j!}} = 2.7182\;81828\;45905} $$

(1:7:5)

or by the limit operation

$$e = \mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n = 2.7182\;81828\;45905$$

(1:7:6)

A limit operation also defines Euler’s constant

$$\gamma = \mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \ln (n)} \right) = \mathop {\lim }\limits_{n \to \infty } \left( {\sum\limits_{j = 1}^n {\frac{1}{j} - \ln (n)} } \right) = 0.57721\;56649\;01533$$

(1:7:7)

The latter is also known as Mascheroni’s constant (Lorenzo Mascheroni, Italian priest, 1750 −1800) and is often denoted by C. Confusingly, authors who employ C to represent Euler’s constant may use γ to represent e ^C.

Also of widespread occurrence throughout the Atlas is the Gauss’s constant

$$g = \frac{1}{{{\rm{mc}}\left( {1,\sqrt 2 } \right)}}\; = \;0.83462\;68416\;74073$$

(1:7:8)

where mc denotes the common, or arithmeticogeometric, mean [Section 61:14]. It is related to the ubiquitous constant U through Ug = $1/\sqrt 2$. Other named constants are Apéry’s constant Z [Section 3:7] and the golden section υ [Section 23:14].

A very important family of constants are the integers, …,−3,−2,−1,0,1,2,3,… and especially the natural numbers, 1,2,3,… discussed in Section 1:14. Other families that occur principally in coefficients of series expansions are the factorials n! [Chapter 2], Bernoulli numbers B _n [Chapter 4], and Euler numbers E _n [Chapter 5]. Fibonacci numbers are discussed in Section 23:14.

1.8 Numerical Values

Equator provides values of the constants π, G, e, γ, g, Z, and υ, exact to 15 digits. Simply type the corresponding keyword, which is pi, catalan, ebase, euler, gauss, apery, or golden. These keywords may be freely used in “constructing” the variable(s) of any other Equator function, as explained in Appendix C. As well as these seven mathematical constants, many physical constants are available through Equator: see Appendix A for these.

1.9 Limits And Approximations

Approximations are seldom needed for constants, but approximations as fractions are available through Equator’s rational approximation routine (keyword rational) [Section 8:13].

1.10 Operations Of The Calculus

Differentiation gives

$$\frac{{\rm{d}}}{{{\rm{d}}x}}c = 0$$

(1:10:1)

while indefinite and definite integration produce

$$\int\limits_0^x {\;c\;{\rm{d}}t} = cx$$

(1:10:2)

and

$$\int\limits_{x_0 }^{x_1 } {c\;{\rm{d}}t} = c\left( {x_1 - x_0 } \right)$$

(1:10:3)

respectively. The result

$$\int\limits_0^\infty {\;c\;\exp ( - st)\;{\rm{d}}t} = \{ c\} = \frac{c}{s}$$

(1:10:4)

describes the Laplace transformation of a constant.

The results of semidifferentiation and semiintegration [Section 12:14] with a lower limit of zero are

$$\frac{{{\rm{d}}^{{\raise0.5ex\hbox{$\scriptstyle {\rm{1}}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {\rm{2}}$}}} }}{{{\rm{d}}x^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}c = \frac{c}{{\sqrt {\pi x} }}$$

(1:10:5)

and

$$\frac{{{\rm{d}}^{ - {\raise0.5ex\hbox{$\scriptstyle {\rm{1}}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle {\rm{2}}$}}} }}{{{\rm{d}}x^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}c = 2c\sqrt {\frac{x}{\pi }} $$

(1:10:6)

Differintegration [Section 12:14] with a lower limit of zero yields

$$\frac{{{\rm{d}}^v }}{{{\rm{d}}x^v }}c = \frac{{cx^{ - v} }}{{\Gamma (1 - v)}}$$

(1:10:7)

where Γ is the gamma function [Chapter 43]. In fact, equations 1:10:1, 1:10:2, 1:10:5, and 1:10:6 are the v = 1, −1, ½ and −½ instances of 1:10:7.

1.11 Complex Argument

A complex constant can be expressed in terms of two real constants in either rectangular or polar notation

$$c = \left\{ {\begin{array}{lllll} \upalpha + i\upbeta & {\rm{where}} & \upalpha = \rho \cos (\theta ) & {\rm{and}} & \upbeta = \rho \sin (\theta )\\ \rho \exp (i\theta ) & {\rm{where}} & \rho = \sqrt {\upalpha ^2 + \upbeta ^2 } & {\mathop{\rm and}\nolimits} & \theta = \arctan (\upbeta /\upalpha ) + \pi \left[ {1 - {\mathop{\rm sgn}} (\upalpha )]/2} \right]\\ \end{array}} \right.$$

(1:11:1)

with i = $\sqrt { - 1}$. The names real part, imaginary part, modulus, and phase are accorded to α, β, ρ, and θ. Figure 1-2 shows how α, β, ρ and θ are related. The expression c = α + iβ is the more useful in formulating the rules for the addition or subtraction of two complex constants:

$$c_1 \pm c_2 = (\upalpha _1 + i\upbeta _1 ) \pm (\upalpha _2 + i\upbeta _2 ) = (\upalpha _1 \pm \upalpha _2 ) + i(\upbeta _1 \pm \upbeta _2 )$$

(1:11:2)

Figure 1-2

Figure 1-2

whereas c = ρexp(iθ) is the more convenient to formulate the multiplication

$$c_1 c_2 = [\rho _1 \exp (i\theta _1 )][\rho _2 \exp (i\theta _2 )] = \rho _1 \rho _2 \exp \left\{ {i(\theta _1 + \theta _2 )} \right\}$$

(1:11:3)

or division

$$\frac{{c_1 }}{{c_2 }} = \frac{{\rho _1 \exp (i\theta _1 )}}{{\rho _2 \exp (i\theta _2 )}} = \frac{{\rho _1 }}{{\rho _2 }}\exp \left\{ {i(\theta _1 - \theta _2 )} \right\}$$

(1:11:4)

of two complex numbers, or in the raising of a complex number to a real power

$$c^v = [\rho \exp (i\theta )]^v = \rho ^v \exp (iv\theta )$$

(1:11:5)

If v is not an integer, this exponentiation operation gives rise to a multivalued complex number [see, for example, Section 13:14]. The raising of a real number to a complex-valued power is handled by the expression

$$v^{\upalpha + i\upbeta } = v^\upalpha \exp \{ i\upbeta \ln (v)\} $$

(1:11:6)

provided that v is positive.

The inverse Laplace transform of the constant c is a Dirac function [Chapter 9], of magnitude c, located at the origin

$$\int\limits_{\upalpha - i\infty }^{\upalpha + i\infty } {c\;\frac{{\exp \left( {ts} \right)}}{{2\pi i}}{\rm{d}}s = {\sc I} \{ c\} } = c\delta (t)$$

(1:11:7)

1.12 Generalizations

A constant is a member of the polynomial function family, other members of which are discussed in Chapters 19−25. The constant function is the special b = 0 case of the linear function discussed in Chapter 7.

1.13 Cognate Functions

Whereas the constant function has the same value for all x, the related pulse function is zero at values of the argument outside a “window” of width h, and is a nonzero constant, c, within this window. The concept of a general “window function” is discussed in Section 9:13. The pulse function in Figure 1-3 takes the value c in the range a−(h/2) < x < a+(h/2) but equals zero elsewhere. The value of the a parameter establishes the location of the pulse, while c and h are termed the pulse height and pulse width respectively. The pulse function may be represented by

$$c\left[ {{\rm{u}}\left( {x - a + \frac{h}{2}} \right) - {\rm{u}}\left( {x - a - \frac{h}{2}} \right)} \right]$$

(1:13:1)

Figure 1-3

Figure 1-3

in terms of the Heaviside function [Chapter 9].

The addition of a number of pulse functions, having various locations, heights, and widths, produces a function whose map consists of horizontal straight line segments. Such a function, known as a piecewise-constant function, may be used to approximate a more complicated or incompletely known function. It is the approximation recorded, for example, whenever a varying quantity is measured by a digital instrument.

1.14 Related Topic: The Natural Numbers

The natural numbers, 1,2,3,… are ubiquitous in mathematics and science. We record here several results for finite sums of their powers:

$$1 + 2 + 3 + \cdots + n = \sum\limits_{j = 1}^n j = \frac{{n(n + 1)}}{2}\quad \quad \quad n = 1,2,3, \cdots $$

(1:14:1)

$$1^2 + 2^2 + 3^2 + \cdots + n^2 = \sum\limits_{j = 1}^n {j^2 } = \frac{{n(n + 1)(2n + 1)}}{6}\quad \quad \quad n = 1,2,3, \cdots $$

(1:14:2)

$$1^3 + 2^3 + 3^3 + \cdots + n^3 = \sum\limits_{j = 1}^n {j^3 } = \frac{{n^2 (n + 1)^2 }}{4}\quad \quad \quad n = 1,2,3, \cdots $$

(1:14:3)

Similarly, the sums of fourth and fifth powers of the first n natural numbers are n(n+1)(2n+1)(3n ²+3n−1)/30 and n ²(n+1)²(2n ²+2n−1)/12, respectively. The general case is

$$1^m + 2^m + 3^m + \cdots + n^m = \sum\limits_{j = 1}^n {j^m } = \frac{{{\rm{B}}_{m + 1} (n + 1) - {\rm{B}}_{m + 1} }}{{m + 1}}\quad \quad \quad n,m = 1,2,3, \cdots $$

(1:14:4)

where B _m denotes a Bernoulli number [Chapter 4] and B _m (x) denotes a Bernoulli polynomial [Chapter 19]. If m is not an integer, summation 1:14:4 may be evaluated generally by equation 12:5:5. The sum of the reciprocals of the first n natural numbers is

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \sum\limits_{j = 1}^n {\frac{1}{j}} = \gamma + \psi (n + 1)\quad \quad \quad n = 1,2,3, \cdots $$

(1:14:5)

where γ is Euler’s constant [Section 1:7] and ψ(x) denotes the digamma function [Chapter 44]. When continued indefinitely, the sum 1:14:5 defines the divergent harmonic series.

The corresponding expressions when the signs alternate are

$$1 - 2 + 3 - 4 + \cdots \pm n = - \sum\limits_{j = 1}^n {( - )^j j} = \left\{ {\begin{array}{ll} (n + 1)/2 & n = 1,3,5, \cdots \\ - n/2 & n = 2,4,6, \cdots \\ \end{array}} \right.$$

(1:14:6)

$$1^2 - 2^2 + 3^2 - 4^2 + \cdots \pm n^2 = - \sum\limits_{j = 1}^n {( - )^j j^2 } = \left\{ {\begin{array}{ll} n(n + 1)/2 & n = 1,3,5, \cdots \\ - n(n + 1)/2 & n = 2,4,6, \cdots \\ \end{array}} \right.$$

(1:14:7)

$$1^3 - 2^3 + 3^3 - 4^3 + \cdots \pm n^3 = - \sum\limits_{j = 1}^n {( - )^j j^3 } = \left\{ {\begin{array}{ll} (2n^3 + 3n^2 - 1)/4 & n = 1,3,5, \cdots \\ - n^2 (2n + 3)/4 & n = 2,4,6, \cdots \\ \end{array}} \right.$$

(1:14:8)

$$1^m - 2^m + 3^m - 4^m + \cdots \pm n^m = - \sum\limits_{j = 1}^n {( - )^j j^m } = - \frac{{{\rm{E}}_m (0)}}{2} - \frac{{( - )^n {\rm{E}}_m (n + 1)}}{2}\quad \quad \quad n,m = 1,2,3, \cdots $$

(1:14:9)

where E _m (x) denotes an Euler polynomial [Chapter 20], and

$$\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \pm \frac{1}{n} = - \sum\limits_{j = 1}^n {\frac{{( - 1)^j }}{j}} = \left\{ {\begin{array}{ll} \psi (n + 1) - \psi \left( {\frac{{n + 1}}{2}} \right) & n = 1,3,5, \cdots \\ \psi (n + 1) - \psi \left( {\frac{n}{2} + 1} \right) & n = 2,4,6, \cdots \\ \end{array}} \right.$$

(1:14:10)

Note that, whereas the n = ∞ version of the harmonic series 1:14:5 does not converge, series 1:14:10 approaches the limit ln(2) as n → ∞.

The numbers 2,4,6,… are called the even numbers. Sums of their powers are easily found by using the identity

$$2^m + 4^m + 6^m + \cdots + n^m = 2^m \left[ {1^m + 2^m + 3^m + \cdots + \left( {\frac{n}{2}} \right)^m } \right]\quad \quad \quad n = 2,4,6, \cdots $$

(1:14:11)

in conjunction with equations 1:14:1−1:14:5. Likewise, use of these equations, together with the identity

$$1^m + 3^m + 5^m + \cdots + n^m = \left[ {1^m + 2^m + 3^m + \cdots + n^m } \right] - 2^m \left[ {1^m + 2^m + 3^m + \cdots + \left( {\frac{{n - 1}}{2}} \right)^m } \right]\;\quad n = 1,3,5, \cdots $$

(1:14:12)

permits sums of powers of the odd numbers, 1,3,5,…, to be evaluated.

For the infinite sums $\sum j^{ - v}$ where j runs from 1 to ∞, see Chapter 3. The same chapter also addresses the related infinite sums $\sum ( - )^j j^{ - v}$, $\sum (2j - 1)^{ - v}$, and $\sum ( - )^j (2j - 1)^{ - v}$. For other sums of numerical series, see Sections 44:14 and 64:6.