The Linear Function bx + c and Its Reciprocal

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Many relationships in science and engineering take the form f(x) = bx + c, and many others can be cast into this form by a redefinition of the variables. The analysis of experimental results can thereby often be reduced to the determination of the coefficients b and c from paired x, f(x) data. The linear regression or least squares method of performing this analysis is exposed in Section 7:14.

7.1 Notation

The linear function bx + c, or its special case 1+ x, is sometimes referred to as a “binomial function” but confusingly it is also termed a “monomial function”. Neither name is used in this Atlas.

The constant c is termed the intercept, whereas b is known as the slope or gradient. These names derive, of course, from the observation that, if the linear function bx + c is plotted versus x, its graph is a straight line that intersects the vertical axis at altitude c and whose inclination from the horizontal is characterized by the number b, which measures the rate at which the function’s value increases with x. The b coefficient is also the tangent [Chapter 34] of angle θ shown in Figure 7-1. The term “slope” is occasionally associated with the angle θ itself, but it more usually means tan(θ), being negative if θ is an obtuse angle (90^o < θ < 180^o). The letter m often replaces b as a symbol for the slope. In the figures, but not elsewhere in the chapter, b and c are assumed positive.

Figure 7-1

Figure 7-1

The name “inverse linear function” is sometimes given to the function 1/(bx + c). Throughout this Atlas, however, we reserve the phrase “inverse function” for the relationship described in equation 0:3:3. The unambiguous name reciprocal linear function is used here.

The name rectangular hyperbola may also be associated with the function 1/(bx + c). However, this name is also applicable to other functions, described in Section 7:13, that share the same shape as, but possess a different orientation to, the reciprocal linear function illustrated in Figure 7-2.

Figure 7-2

Figure 7-2

7.2 Behavior

The linear function bx + c is defined for all values of the argument x and (unless b = 0) itself assumes all values. The same is true of the reciprocal linear function which, however, displays an infinite discontinuity at x = −c/b, as illustrated in Figure 7-2. A graph of the reciprocal linear function f(x) = 1/(bx + c) has a high degree of symmetry [Section 14:15], being inversion symmetric about the point x = −c/b, f = 0 and displaying mirror symmetry towards reflections in the lines −xsgn(b)−(c/b) and xsgn(b)+(c/b). Here sgn is the signum function [Chapter 8].

7.3 Definitions

The arithmetic operations of multiplication by b and addition of c fully define f(x) = bx + c. The same operations, followed by division into unity, define the reciprocal linear function.

The linear function is completely characterized when its values, f ₁ and f ₂, are known at two (distinct) arguments, x ₁ and x ₂. The slope and intercept may be found from the formula

$$bx + c = \left( {\frac{{f_2 - f_1 }}{{x_2 - x_1 }}} \right)x + \frac{{x_2 f_1 - x_1 f_2 }}{{x_2 - x_1 }}$$

(7:3:1)

7.4 Special Cases

When b = 0, the linear function and its reciprocal reduce to a constant [Chapter 1]. When c = 0, the linear function is proportional to its argument x.

7.5 Intrarelationships

Both the linear function and its reciprocal obey the reflection formula

$${\rm{f}}\left( { - x - \frac{c}{b}} \right) = - {\rm{f}}\left( {x - \frac{c}{b}} \right)\quad \quad \quad {\rm{f}}(x) = bx + c\quad {\rm{or}}\quad \frac{1}{{bx + c}}$$

(7:5:1)

Two linear functions that share the same b parameter represent straight lines that are parallel; the distance separating these lines is $\left| {c_2 - c_1 } \right|/\sqrt {1 + b^2 }$. If the slopes of two straight lines satisfy the relation b ₁ b ₂ = −1, the lines are mutually perpendicular. intersecting at the point x = (c ₂−c ₁)/(b ₁−b ₂). The inverse of the linear function f(x) = bx + c, defined by F{f(x)} = x, is another linear function F(x) = (x/b)−(c/b); graphically, these two straight lines usually cross at x = c(1+b ²)/(1−b ²).

The sum or difference of two linear functions is a third linear function (b ₁±b ₂)x + c ₁ ± c ₂ and this property extends to multiple components. The product of two, three, or many linear functions is a quadratic function [Chapter 15], a cubic function [Chapter 16] or a higher polynomial function [Chapter 17]. Unless b ₂ c ₁ = b ₁ c ₂, the quotient of two linear functions is the infinite power series:

$$\frac{{b_1 x + c_1 }}{{b_2 x + c_2 }} = \left\{ \begin{array}{ll} \frac{{c_1 }}{{c_2 }} + \left( {\frac{{c_1 }}{{c_2 }} - \frac{{b_1 }}{{b_2 }}} \right)\sum\limits_{j = 1}^\infty {\left( {\frac{{ - b_2 x}}{{c_2 }}} \right)}^j & \left| x \right| < \left| {\frac{{c_2 }}{{b_2 }}} \right| \\ \frac{{b_1 }}{{b_2 }} - \left( {\frac{{c_1 }}{{c_2 }} - \frac{{b_1 }}{{b_2 }}} \right)\sum\limits_{j = 1}^\infty {\left( {\frac{{ - c_2 }}{{b_2 x}}} \right)}^j & \left| x \right| > \left| {\frac{{c_2 }}{{b_2 }}} \right| \\ \end{array} \right.$$

(7:5:2)

The sum or difference of two reciprocal linear functions is a rational function [Section 17:13] of numeratorial and denominatorial degrees of 1 and 2 respectively. Finite series of certain reciprocal linear functions may be summed in terms of the digamma function [Chapter 44]

$$\frac{1}{c} + \frac{1}{{x + c}} + \frac{1}{{2x + c}} + \cdots + \frac{1}{{Jx + c}} = \sum\limits_{j = 0}^J {\frac{1}{{jx + c}} = \frac{1}{x}\left[ {\psi \left( {J + 1 + \frac{c}{x}} \right) - \psi \left( {\frac{c}{x}} \right)} \right]} $$

(7:5:3)

or in terms of Bateman’s G function [Section 44:13]

$$\frac{1}{c} - \frac{1}{{x + c}} + \frac{1}{{2x + c}} - \cdots \pm \frac{1}{{Jx + c}} = \sum\limits_{j = 0}^J {\frac{{( - 1)^j }}{{jx + c}} = \frac{1}{{2x}}\left[ {{\rm{G}}\left( {\frac{c}{x}} \right) \pm {\rm{G}}\left( {J + 1 + \frac{c}{x}} \right)} \right]} $$

(7:5:4)

In formula 7:5:4, the upper/lower signs are taken depending on whether J is even or odd. The corresponding infinite sum is

$$\frac{1}{c} - \frac{1}{{x + c}} + \frac{1}{{2x + c}} - \cdots = \sum\limits_{j = 0}^\infty {\frac{{( - 1)^j }}{{jx + c}} = \frac{1}{{2x}}{\rm{G}}\left( {\frac{c}{x}} \right)} $$

(7:5:5)

See Section 44:14 for further information on this topic.

7.6 Expansions

The linear function may be expanded as a infinite series of Bessel functions [Chapter 52]

$$bx + c = c + 2\;{\rm{J}}_1 (bx) + 6\;{\rm{J}}_3 (bx) + 10\;{\rm{J}}_5 (bx) + \cdots = c + 2\sum\limits_{j = 1}^\infty {(2j - 1)} \;{\rm{J}}_{2j - 1} (bx)$$

(7:6:1)

though this representation is seldom employed.

The reciprocal linear function is expansible as a geometric series in alternative forms

$$\frac{1}{{bx + c}} = \left\{ \begin{array}{ll} \frac{1}{c} - \frac{{bx}}{{c^2 }} + \frac{{b^2 x^2 }}{{c^3 }} - \frac{{b^3 x^3 }}{{c^4 }} + \cdots = \frac{1}{c}\sum\limits_{j = 0}^\infty {\left( {\frac{{ - bx}}{c}} \right)^j } & \left| x \right| < \left| {\frac{c}{b}} \right| \\ \frac{1}{{bx}} - \frac{c}{{b^2 x^2 }} + \frac{{c^2 }}{{b^3 x^3 }} - \frac{{c^3 }}{{b^4 x^4 }} + \cdots = \frac{1}{{bx}}\sum\limits_{j = 0}^\infty {\left( {\frac{{ - c}}{{bx}}} \right)^j } & \left| x \right| > \left| {\frac{c}{b}} \right| \\ \end{array} \right.$$

(7:6:2)

according to the magnitude of the argument x compared to that of the ratio c/b. Likewise there are two alternatives when the reciprocal linear function is expanded as an infinite product

$$\frac{1}{{bx + c}} = \left\{ \begin{array}{ll} \frac{{c - bx}}{{c^2 }}\prod\limits_{j = 1}^\infty {\left[ {1 + \left( {\frac{{bx}}{c}} \right)^{2^j } } \right]} & - 1 < \frac{{bx}}{c} < 1 \\ \frac{{bx - c}}{{b^2 x^2 }}\prod\limits_{j = 1}^\infty {\left[ {1 + \left( {\frac{c}{{bx}}} \right)^{2^j } } \right]} & \left| {\frac{{bx}}{c}} \right| > 1 \\ \end{array} \right.$$

(7:6:3)

For example, if |x| < 1

$$\frac{1}{{1 \pm x}} = \left( {1 \mp x} \right)\left( {1 + x^2 } \right)\left( {1 + x^4 } \right)\left( {1 + x^8 } \right)\left( {1 + x^{16} } \right) \cdots $$

(7:6:4)

7.7 Particular Values

As Figure 7-1 shows, the linear function bx + c equals c when x = 0 and equals zero when x = −c/b. The reciprocal linear function has neither an extremum nor a zero, but it incurs a discontinuity at x = −c/b [Figure 7-2].

7.8 Numerical Values

These are easily calculated by direct substitution. The construction feature of Equator enables a linear function to be used as the argument of another function.

7.9 Limits And Approximations

The reciprocal linear function approaches zero asymptotically as x → ±∞.

7.10 Operations Of The Calculus

The rules for differentiation of the linear function and its reciprocal are

$$\frac{{\mathop{\rm d}\nolimits} }{{{\rm{d}}x}}(bx + c) = b$$

(7:10:1)

and

$$\frac{{\mathop{\rm d}\nolimits} }{{{\rm{d}}x}}\left( {\frac{1}{{bx + c}}} \right) = \frac{{ - b}}{{(bx + c)^2 }}$$

(7:10:2)

while those for indefinite integration are

$$\int\limits_{\rm{0}}^x {\;(bt + c){\rm{d}}t} = \frac{{bx^2 }}{2} + cx$$

(7:10:3)

and

$$\int\limits_{\rm{0}}^x {\;\frac{1}{{bt + c}}\;{\rm{d}}t} = \frac{1}{b}\ln \left( {\left| {\frac{{bx + c}}{c}} \right|} \right)$$

(7:10:4)

If 0 < −c/b < x, the integrand in 7:10:4 encounters an infinity; in this event, the integral is to be interpreted as a Cauchy limit. This means that the ordinary definition of the integral is replaced by

$$\mathop {\lim }\limits_{\varepsilon \to 0} \left\{ {\int\limits_0^{ - (c/b) - \varepsilon } {\frac{1}{{bt + c}}\;{\rm{d}}t\; + } \int\limits_{ - (c/b) + \varepsilon }^x {\frac{1}{{bt + c}}\;{\rm{d}}t} } \right\}$$

(7:10:5)

Several other integrals in this section, including those that follow immediately, may also require interpretation as Cauchy limits, but mention of this will not always be made

$$\int\limits_0^x {\frac{1}{{(Bt + C)(bt + c)}}{\rm{d}}t = \frac{1}{{bC - Bc}}\ln \left( {\left| {\frac{{C(bx + c)}}{{c(Bx + C)}}} \right|} \right)} \quad \quad \quad Bc \ne bC$$

(7:10:6)

$$\int\limits_0^x {\frac{{Bt + C}}{{bt + c}}{\rm{d}}t = \frac{{Bx}}{b} + \frac{{bC - Bc}}{{b^2 }}\ln \left( {\left| {\frac{{bx + c}}{c}} \right|} \right)} $$

(7:10:7)

$$\int\limits_0^x {\frac{{t^n }}{{bt + c}}{\mathop{\rm d}\nolimits} x = \frac{{\left( { - c} \right)^n }}{{b^{n + 1} }}\left[ {\ln \left( {\left| {\frac{{bx + c}}{c}} \right|} \right) + \sum\limits_{j = 1}^n {\frac{1}{j}\left( {\frac{{ - bx}}{c}} \right)^j } } \right]\quad \quad \quad n = 0,1,2, \cdots } $$

(7:10:8)

Formulas for the semidifferentiation and semiintegration [Section 12:14] of the linear function are

$$\frac{{{\mathop{\rm d}\nolimits} ^{1/2} }}{{{\rm{d}}x^{1/2} }}(bx + c) = \frac{{2bx + c}}{{\sqrt {\pi x} }}$$

(7:10:9)

and

$$\frac{{{\mathop{\rm d}\nolimits} ^{ - 1/2} }}{{{\rm{d}}x^{ - 1/2} }}(bx + c) = \sqrt {\frac{x}{\pi }} \left[ {\frac{{4bx}}{3} + 2c} \right]$$

(7:10:10)

when the lower limit is zero. The table below shows the semiderivatives and semiintegrals of the reciprocal linear functions 1/(bx+c) and 1/(bx−c), when the lower limit is zero, and when it is −∞. In this table, but not necessarily elsewhere in the chapter, b and c are positive.

	${\mathop{\rm f}\nolimits} (x) = \frac{1}{{bx + c}}$	${\mathop{\rm f}\nolimits} (x) = \frac{1}{{bx - c}}$
$\frac{{{\mathop{\rm d}\nolimits} ^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {\mathop{\rm f}\nolimits} }}{{{\rm{d}}x^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}$	$\frac{{\sqrt {bx + c} - \sqrt {bx} \;{\rm{arsinh}}\left( {\sqrt {bx/c} } \right)}}{{\sqrt {\pi x(bx + c)^3 } }}\quad x > 0$	$\frac{{ - \sqrt {c - bx} - \sqrt {bx} \;{\rm{arcsin}}\left( {\sqrt {bx/c} } \right)}}{{\sqrt {\pi x(c - bx)^3 } }}\quad 0 < x < \frac{c}{b}$
$\frac{{{\mathop{\rm d}\nolimits} ^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {\mathop{\rm f}\nolimits} }}{{{\rm{d}}x^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}$	$\frac{{2{\rm{arsinh}}\left( {\sqrt {bx/c} } \right)}}{{\sqrt {\pi b(bx + c)} }}\quad x > 0$	$\frac{{ - 2{\rm{arcsin}}\left( {\sqrt {bx/c} } \right)}}{{\sqrt {\pi b(c - bx)} }}\quad 0 < x < \frac{c}{b}$
$\left. {\frac{{{\mathop{\rm d}\nolimits} ^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}{{{\rm{d}}t^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}{\mathop{\rm f}\nolimits} (t)} \right|_{ - \infty }^x$	$\frac{1}{2}\sqrt {\frac{{\pi b}}{{( - bx - c)^3 }}} \quad x < \frac{{ - c}}{b}$	$\frac{{ - 1}}{2}\sqrt {\frac{{\pi b}}{{(c - bx)^3 }}} \quad x < \frac{c}{b}$
$\left. {\frac{{{\mathop{\rm d}\nolimits} ^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}{{{\rm{d}}t^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} }}{\mathop{\rm f}\nolimits} (t)} \right|_{ - \infty }^x$	$- \sqrt {\frac{\pi }{{b( - bx - c)}}} \quad x < \frac{{ - c}}{b}$	$- \sqrt {\frac{\pi }{{b(c - bx)}}} \quad x < \frac{c}{b}$

See Sections 12:14 and 64:14 for the definitions and symbolism of semidifferintegrals with various lower limits.

The Laplace transforms of the linear and reciprocal linear functions are

$$\int\limits_0^\infty {(bt + c)\exp ( - st)\;{\rm{d}}t = {\sc l} \left\{ {bt + c} \right\} = \frac{{cs + b}}{{s^2 }}} $$

(7:10:11)

and

$$\int\limits_0^\infty {\frac{1}{{bt + c}}\exp ( - st){\rm{d}}t = {\sc l} \left\{ {\frac{1}{{bt + c}}} \right\} = \frac{{ - 1}}{b}\exp \left( {\frac{{cs}}{b}} \right){\rm{Ei}}\left( {\frac{{ - cs}}{b}} \right)} $$

(7:10:12)

the latter involving an exponential integral [Chapter 37]. A general rule for the Laplace transformation of the product of any transformable function f(t) and a linear function is

$$\int\limits_0^\infty {(bt + c){\mathop{\rm f}\nolimits} \;(t)\exp ( - st){\rm{d}}t = {\sc l} \left\{ {(bt + c){\rm{f}}(t)} \right\} = c {\sc l} \left\{ {{\rm{f}}(t)} \right\} - b\frac{{\mathop{\rm d}\nolimits} }{{{\rm{d}}s}}\left\{ {{\rm{f}}(t)} \right\}} $$

(7:10:13)

Requiring a Cauchy-limit interpretation, the integral transform

$$\frac{1}{\pi }\int\limits_{ - \infty }^\infty {{\rm{f}}(t)\frac{{{\rm{d}}t}}{{t - y}}} $$

(7:10:14)

is called a Hilbert transform (David Hilbert, German mathematician, 1862 −1943). The Hilbert transforms of many functions, mostly piecewise-defined functions [Section 8:4], are tabulated by Erdélyi, Magnus, Oberhettinger and Tricomi [Tables of Integral Transforms, Volume 2, Chapter 15]. For example, the Hilbert transform of the pulse function [Section 1:13] is

$$\frac{1}{\pi }\int\limits_{ - \infty }^\infty {c\left[ {u\left( {t - a + \frac{h}{2}} \right) - u\left( {t - a - \frac{h}{2}} \right)} \right]\frac{{{\rm{d}}t}}{{t - y}}} = \frac{c}{\pi }\ln \left( {\frac{{2(a - y) + h}}{{2(a - y) - h}}} \right)$$

(7:10:15)

A valuable feature of Hilbert transformation is that the inverse transform is identical in form to the forward transformation, apart from a sign change.

7.11 Complex Argument

The linear function of complex argument, and its reciprocal, split into the real and imaginary parts

$$bz + c = (bx + c) + iby$$

(7:11:1)

and

$$\frac{1}{{bz + c}} = \frac{{bx + c}}{{(bx + c)^2 + b^2 y^2 }} - i\frac{{by}}{{(bx + c)^2 + b^2 y^2 }}$$

(7:11:2)

if b and c are real.

The inverse Laplace transformation of the linear and reciprocal linear functions leads to functions from Chapters 10 and 26

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

(7:11:3)

and

$$\int\limits_{\alpha - i\infty }^{\alpha + i\infty } {\frac{1}{{bs + c}}\frac{{\exp (ts)}}{{2\pi i}}\;{\rm{d}}s = {\sc I} \left\{ {\frac{1}{{bs + c}}} \right\} = \frac{1}{b}\exp \left( {\frac{{ - ct}}{b}} \right)} $$

(7:11:4)

The Laplace inversion of a function of bx + c is related to the inverse Laplace transform of the function itself through the general formula

$$\int\limits_{\alpha - i\infty }^{\alpha + i\infty } {{\mathop{\rm f}\nolimits} \;(bs + c)\frac{{\exp (ts)}}{{2\pi i}}\;{\rm{d}}s = {\sc I} \left\{ {{\rm{f}}(bs + c)} \right\} = \frac{1}{b}\exp \left( {\frac{{ - ct}}{b}} \right)I\left\{ {{\rm{\bar f}}\;{\rm{(}}s{\rm{)}}} \right\}} $$

(7:11:5)

7.12 Generalizations

The linear and reciprocal linear functions are the v = ±1 cases of the more general function (bx + c)^v, two other instances of which are addressed in Chapter 11. More broadly, all the functions of Chapters 10−14 are particular examples of a wide class of algebraic functions generalized by the formula (bx ⁿ + c)^v.

The linear function is an early member of a hierarchy in which the quadratic and cubic functions [Chapters 15 and 16] are higher members and which generalizes to the polynomial functions discussed Chapter 17. All the “named” polynomials [Chapters 18−24] have a linear function as one member of their families.

7.13 Cognate Functions

A frequent need in science and engineering is to approximate a function whose values, f ₀ , f ₁ , f ₂ , ··· , f _n , are known only at a limited number of arguments x ₀, x ₁, x ₂, ···, x _n , the so-called data points. The simplest way of constructing a function that fits all the known data, but one that is adequate in many applications, is by using a piecewise-linear function. In graphical terms, this implies simply “connecting the dots”. For any argument lying between two adjacent data points, the interpolation

$${\rm{f}}(x) = \frac{{\left( {x_{j + 1} - x} \right)f_j + \left( {x - x_j } \right)f_{j + 1} }}{{x_{j + 1} - x_j }}\quad \quad \quad x_j \le x \le x_{j + 1} $$

(7:13:1)

applies. Usually a piecewise-linear function has discontinuities at all the interior data points. This defect is overcome by the “sliding cubic” and “cubic spline” interpolations exposed in Section 17:14.

The simplest reciprocal linear functions 1/(1 ± x) serve as prototypes, or basis functions, for all L = K hypergeometric functions [Section 18:14]. All the functions in Tables 18-1 and 18-2, as well as many others, may be “synthesized” [Section 43:14] from 1/(1+ x) or 1/(1− x).

The reciprocal linear function is related to the function addressed in Section 15:4 because the shape of each is a rectangular hyperbola. Thus, clockwise rotation [Section 14:15] through an angle of π/4 of the curve 1/(bx + c) about the point x = −c/b on the x-axis produces a new function,

$$\pm \sqrt {\left( {x + \frac{c}{b}} \right)^2 + \frac{2}{b}} $$

(7:13:2)

that is a rectangular hyperbola of the class discussed in Section 15:4.

7.14 RELATED TOPIC: linear regression

Frequently experimenters collect data that are known, or believed, to obey the equation f = f(x) = bx + c but which incorporate errors. From the data, which consists of the n pairs of numbers (x ₁ , f ₁), (x ₂ , f ₂), (x ₃ , f ₃), ···, (x _n , f _n ), the scientist needs to find the b and c coefficients of the best straight line through the data, as in Figure 7-3. If the errors obey, or are assumed to obey, a Gaussian distribution [Section 27:14] and are entirely associated with the measurement of f (that is, the x values are exact), then the adjective “best” implies minimizing the sum of the squared deviations, $\sum {(bx + c - f)^2 }$. The procedure for finding the coefficients that achieve this minimization is known as linear regression or least squares and leads to the formulas

$$b = \frac{{n\sum {xf} - \sum x \sum f }}{{n\sum {x^2 } - \left( {\sum x } \right)^2 }} = \frac{{6\left[ {2\sum {jf - (n + 1)\sum f } } \right]}}{{n(n^2 - 1)h}}$$

(7:14:1)

and

$$c = \frac{{\sum {x^2 } \sum f - \sum x \sum {xf} }}{{n\sum {x^2 } - \left( {\sum x } \right)^2 }} = \frac{{\sum f - b\sum x }}{n} = \frac{{\sum f }}{n} - \frac{{b(x_1 + x_n )}}{2}\;$$

(7:14:2)

Figure 7-3

Figure 7-3

An abbreviated notation, exemplified by

$$\sum {xf = \sum\limits_{j = 1}^n {x_j f_j } } $$

(7:14:3)

is used in the formulas of this section. Evaluation of these formulas simplifies considerably in the common circumstance in which data are gathered with equal spacing, this is when x ₂−x ₁ = x ₃−x ₂ = ··· = x _n −x _{n − 1} = h. The simplified formulas appear in red in equations 7:14:1, 7:14:2, 7:14:4 and 7:14:8.

A measure of how well the data obey the linear relationship is provided by the correlation coefficient, given by

$$r = \frac{{n\sum {xf} - \sum x \sum f }}{{\sqrt {\left[ {n\sum {x^2 } - \left( {\sum x } \right)^2 } \right]\left[ {n\sum {f^2 } - \left( {\sum f } \right)^2 } \right]} }} = b\sqrt {\frac{{n\sum {x^2 } - \left( {\sum x } \right)^2 }}{{n\sum {f^2 } - \left( {\sum f } \right)^2 }}} = \frac{{nhb}}{6}\sqrt {\frac{{3(n^2 - 1)}}{{n\sum {f^2 } - \left( {\sum f } \right)^2 }}} $$

(7:14:4)

Values close to ±1 imply a good fit of the data to the linear function, whereas r will be close to zero if there is little or no correlation between f and x. Sometimes r ² is cited instead of r.

Commonly there is a need to know not only what the best values are of the slope b and the intercept c but also what uncertainties attach to these best values. Quoting their standard errors [Section 40:14] in the format

$${\rm{slope}} = b \pm \Delta b\quad \quad \quad {\rm{where}}\quad \Delta b = {\text{standard error in }}b$$

(7:14:5)

and

$${\rm{intercept}} = c \pm \Delta c\quad \quad \quad {\rm{where}}\quad \Delta c = {\text{standard error in }}c$$

(7:14:6)

is a succinct way of reporting the uncertainties associated with least squares determinations. the significance to be attached to these statements is that the probability is approximately 68% that the true slope will lie between b −Δb and b + Δb. Similarly, there is a 68% probability that the true intercept lies in the range c ± Δc. The formulas giving these standard errors are

$$\Delta b = \sqrt {\;\frac{{n\sum {f^2 } - \left( {\sum f } \right)^2 }}{{n\sum {x^2 } - \left( {\sum x } \right)^2 }} - \left[ {\frac{{n\sum {xf} - \sum {x\sum f } }}{{n\sum {x^2 } - \left( {\sum x } \right)^2 }}} \right]^2 } = b\sqrt {\frac{1}{{r^2 }} - 1} $$

(7:14:7)

and

$$\Delta c = \Delta b\sqrt {\frac{{\sum {x^2 } }}{n}} = \frac{{\Delta b}}{2}\sqrt {\left( {x_1 + x_n } \right)^2 + \frac{{(n^2 - 1)h^2 }}{3}} $$

(7:14:8)

A related, but simpler, problem is the construction of the best straight line through the points (x ₁ , f ₁), (x ₂ , f ₂), (x ₃ , f ₃), ···, (x _n , f _n ), with the added constraint that the line must pass through the point (x ₀, f ₀). In practical problems this obligatory point is often the x = 0, f = 0 origin. Equations 7:14:1 and 7:14:2 should not be used in these circumstances, though they often are. The appropriate replacements are

$$b = \frac{{\sum {(x - x_0 )(f - f_0 )} }}{{\sum {(x - x_0 )^2 } }}$$

(7:14:9)

and

$$c = f_0 - bx_0 $$

(7:14:10)

Equations 7:14:1−7:14:8 are based on the assumption that all data points are known with equal reliability, a condition that is not always valid. Variable reliability can be treated by assigning different weights to the points. If the value f ₁ is more reliable than f ₂, then a larger weight w ₁ is assigned to the data pair (x ₁ , f ₁) than the weight w ₂ assigned to point (x ₂ , f ₂). The weights then appear as multipliers in all summations, leading to the formulas

$$b = \frac{{\sum w \sum {wxf - \sum {wx} \sum {wf} } }}{{\sum w \sum {wx} ^2 - \left( {\sum {wx} } \right)^2 }}$$

(7:14:11)

and

$$c = \frac{{\sum {wf - b\sum {wx} } }}{{\sum w }}$$

(7:14:12)

for the slope and intercept. Only the relative weights are of import; the absolute values of w ₁, w ₂, w ₃, ···, w _n have no significance beyond this. In practice, one attempts to assign a weight w _j to the jth point that is inversely proportional to the square of the uncertainty in f _j . Notice that formulas 7:14:1 and 7:14:2 are the special cases of 7:14:11 and 7:14:12 in which all w’s are equal. Similarly, the formulas in 7:14:9 and 7:14:10 result from setting the weight of one point, (x ₀ , f ₀), to be overwhelmingly greater than all the other weights, which are uniform.