## A Generalization of the Exponential Function to

## Model Growth

### Martin Ricker and Dietrich von Rosen

*Abstract*—We generalize the exponential function to model
instantaneous relative growth. The modified function is
defined by a linear relationship between a continuous quantity
(rather than time) and logarithmic relative growth. The
corresponding formula is ln_{}*q*

###

*t*

*q*

###

*t*

_{}

*a b q t*

###

, where###

*q t q t* is instantaneous relative growth of a quantity *q*,
*t*_{ refers to time, }*a* denotes initial logarithmic relative growth,
and *b* is a shape parameter in terms of its sign, as well as a
scaling parameter in terms of its magnitude. For calculating

###

*q t* , the exponential integral Ei

###

*b q*

###

_{}

###

exp###

*b*

*q q q*

###

###

d is needed. The problem of taking the inverse of Ei###

*x*

*z*is addressed. In order to distinguish two possible solutions for given

_{Ei}*z*, we define the two inverse functions Ei( 1)

_{Ei}*x*

_{0}

*zEi*

and ( 1)

0
Ei*x* *zEi*

. An indirect method for their numerical
evaluation is developed. With the generalized exponential
function, one can model sigmoid growth (*b*0), exponential
growth (*b*0), and explosive growth (*b*0), where the term
“explosive growth” refers to a relative growth rate that
increases with time. The resulting formula of generalized
exponential growth is

###

###

###

###

###

###

###

###

###

###

###

( 1) 0 ( 1) 0

1

Ei exp +Ei for 0,

1

Ei exp +Ei for 0,

*C* *C*

*C* *C*

*x*

*x*

*t* *t* *a* *b q* *b*

*b*
*q t*

*t* *t* *a* *b q* *b*

*b*

_{} _{}

_{} _{} _{} _{} _{} _{ } _{} _{}

where *q _{C}* is a calibrating quantity at time

*t*. For

_{C}*b*0, the two functions equal the (standard) exponential function

###

*C*exp

###

*C*

###

exp###

*q t* *q* _{}*t**t* *a*_{}. In the case of sigmoid growth,
the inflection point quantity is 1 /*b*, which depends only on
one parameter (*b*). Negative growth can be modeled by
substituting *t**tC* with *tC**t*. Any two points of logarithmic

relative growth can be connected unambiguously with the

Manuscript received on September 22, 2017; revised on April 18, 2018. The Dirección General de Asuntos del Personal Académico (DGAPA) of the Universidad Nacional Autónoma de México (UNAM) provided a stipend for the first author (MR), to spend a sabbatical year at the Swedish University of Agricultural Sciences in 2014-15.

M. Ricker is with the Instituto de Biología, Universidad Nacional Autónoma de México (UNAM), Mexico City (e-mail: mricker@ib.unam.mx, martin_tuxtlas@yahoo.com.mx).

D. von Rosen is with the Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden, as well as with the Department of Mathematics, Linköping University, Linköping, Sweden (e-mail: Dietrich.von.Rosen@slu.se).

generalized exponential function, to derive the corresponding
function of *q t*

###

. Furthermore, we derive formulas for the conversion of a segmented curve of logarithmic relative growth as a function of time, into an equivalent growth curve of*q t*

###

. Finally, the generalized exponential function is compared with a 2nd_{-degree polynomial and the nonlinear Schnute function. In }conclusion, the generalized exponential function is useful for modeling a path of changing relative growth continuously, and to translate it into a growth curve of quantity as a function of time.

*Index terms - Exponential growth, Exponential function, *
Exponential integral Ei

###

*x*, Growth curves, Inverse of the exponential integral function, Explosive growth, Schnute function, Sigmoid growth

I. INTRODUCTION

Growth curves are monotonically increasing functions over time. In the case of negative growth, they can also be monotonically decreasing functions. Growth curves are of interest in a variety of fields, such as biology, medicine, engineering, or economics [1]. In forest science, there is a long history of applying statistical growth curve models to predict tree size and yield (e.g., [2-3]). To model various types of growth processes, the main aim of our article is to study a generalization of the exponential function [4]. This generalization is constructive from a mathematical point of view, although we are also interested in subsequent statistical applications of tree growth data or other empirical growth phenomena. Here, the non-statistical, mathematical aspects are treated.

Let *q t*

###

represent growth as a function of time, and let###

*q t* denote *dq dt*. Nota that the term “relative growth”
will always refer to instantaneous relative growth

###

.*q t* *q t* The basic characteristic of the exponential
function is that relative growth satisfies

###

exp ,*q t q t =* _{}*c t* _{} (1)
where *c* is a constant. Now we modify this model:

###

exp###

,*q t* *q t* _{}*a b q t* _{} (2)
where the term *c t* has been substituted with *a* *b q*, with

*a* and *b* being constants. The original motivation for this
substitution was to establish a linear relationship with

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

logarithmic relative growth of the form

###

###

ln_{}*q t* *q t* _{}*a* *b q t* , when time *t* is unknown [5].
However, it turns out that the resulting model is interesting
also when time *is* known. It will be shown that the growth
function *q t*

###

can be sigmoid, exponential, or explosive, where the term “explosive” refers to a relative growth rate that increases with time. Furthermore, the resulting growth function has some interesting mathematical properties.The article starts with the derivation of the growth curve function. Then, the numerical calculation of the inverse of the exponential integral function Ei

###

*x*is treated, and the final formulas of the resulting growth function are presented. The inflection point in sigmoid growth, and the relationship among the parameters are discussed. Negative growth is also treated. Subsequently, the new function is compared with that of standard exponential growth. Furthermore, a procedure is derived for the conversion of a segmented curve of logarithmic relative growth as a function of time, into a growth curve of quantity as a function of time. Finally, the new function is compared with a polynomial function and the nonlinear Schnute function.

II. DERIVATION OF THE GROWTH CURVE FUNCTION

Since *q t*

###

is a monotone function, we can first consider###

*t q* instead of *q t*

###

. Write (2) as###

d*q*d

*t*

###

*q*

###

###

exp *a b q* and transform it into d d*t* *q*

###

###

###

###

1 *q*exp *a b q* . Let d

*t*

###

*q*d

*q*

*t*

###

*q*: given

*q*0, we can write

###

exp*a b q*

*t q*

*q*

. (3)

Integrating both sides of this equation with respect to *q*, and
letting *qC* at *tC* be a calibrating point on the growth curve,
yields

###

exp

exp d for 0,

1

exp d for 0.

*C*

*C*
*C*

*C*

*q*
*q*

*q*
*q*

*b v*

*t* *a* *v* *b*

*v*
*t q*

*t* *a* *v* *b*

*v*

_{} _{}

_{} _{}

###

###

Note that 1d*v*
*v*

###

equals ln###

*v*. The exponential integral

###

###

exp d

*b v*
*v*
*v*

###

, however, represents an infinite series, and is denoted Ei_{}

*b v*

_{}

(page 176 in [6]):

###

###

###

###

1 2

Ei ln ...

1! 1 2! 2

*b v* *b v*

*b v*

###

*b v*

, (4)

where | *b v*| refers to the absolute value of *b v*, and !*i*
denotes the factorial of *i*. Furthermore, the
Euler-Mascheroni constant

1 1

lim ln

*n*
*n*

*k*

*n*
*k*

###

_{ }

_{}

0.577215...has been added in (4), so that (4) represents the Cauchy principal value of Ei

###

*x*. The reason is that Ei

###

*x*has a singularity at

*x*0 (Figure 1 top). Whereas in a definite integral cancels out, adding provides a theoretical basis for taking the integral over

*x*0. The exponential integral Ei

_{ }

*x*has been discussed mathematically in detail in [7]. Note that Ei[0] , but

(exp[0*v*] / )d*v v* (1 / )d*v v*

###

###

ln[ ]*v*.

The integral from *qC* to *q* can be calculated numerically
as Ei

###

*b q*

###

Ei###

*b q*

_{C}###

. For example, in the program*Mathematica*(www.wolfram.com/mathematica/), one can use the function “ExpIntegralEi”. Ricker and del Río (Appendix 3 in [5]) also derived a relatively easy-to-implement procedure to calculate the integral up to a desired accuracy, with a remainder

*R x i*

###

,###

:

###

###

###

1 2

Ei ln ...

1! 1 2! 2 ! , ,

*i*

*x* *x* *x*

*x* *x*

*i i*
*R x i*

###

(5a)

where *i* refers to a chosen integer number. For *i* *x*2:

###

###

###

###

1 2

, .

! 1 1

2
*i*

*x*
*R x i*

*x*
*i* *i*

*i*

(5b)

For an interpretation of Ei

_{}

*b q*

_{}

Ei_{}

*b q*

_{C}_{}

, see pages
5-6 in [8].
Thus the time to grow from *qC*0 (at *tC*) to *q*0

###

*q*

*qC*

###

can be calculated as###

Ei Ei

for 0, exp

ln ln

for 0. exp

*C*
*C*

*C*
*C*

*b q* *b q*

*t* *b*

*a*
*t q*

*q* *q*

*t* * b*

*a*

_{} _{} _{} _{}

_{ }

(6)

For *q**qC* the resulting time period *t**tC* is negative. From
(6), the initial logarithmic relative growth rate is derived:

Ei[ ] Ei[ ]

ln for 0,

ln[ ] ln[ ]

ln for 0.

*C*
*C*
*C*
*C*

*b q* *b q*

*b*
*t t*

*a*

*q* *q*

* b*
*t t*

(7)

Since *x* in Ei

_{ }

*x*or ln

_{ }

*x*has to be positive (for non-complex solutions), one gets the following restrictions for (7):

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

*q**q _{C}* for

*t*

*t*, and

_{C}*q*

*q*for

_{C}*t*

*t*. (8) The restriction of (8) is derived in the following way: With

_{C}*C*

*t**t* in the case of *b*0 in (7), one needs

###

###

###

###

Ei *b q* Ei *b q _{C}* . Using the notation shown in Figure
1, for

*b*0 and positive

*q*, this translates into:

###

###

###

###

0 0

Ei_{x}_{} *b q* Ei_{x}_{} *b q _{C}*

*b q*

*b q*

_{C}*q*

*q*. On the other hand, for

_{C}*b*0 and positive

*q*, one gets:

###

###

###

###

0 0

Ei_{x}_{} *b q* Ei_{x}_{} *b q _{C}*

*b q*

*b q*

_{C}*q*

*q*. The reason for

_{C}*b q*

*b qC*in the latter case (rather than

*C*

*b q* *b q*

) is that Ei_{x}_{}_{0}

###

*x*decreases with increasing .

*x* Therefore, for both *b*0 and *b*0 one gets the
condition *q**qC*. For *t**tC*, the inverse logic applies.

For *b*0, the parameter *a* refers only to the *initial*
logarithmic relative growth, i.e., infinitesimally close to

zero:

###

0 lim ln

*q*

*a* *q t* *q*

_{} _{} _{}_{} (see Figure 3 bottom). For the
logarithmic relative growth at *any* quantity *q*0, we
introduce the variable *y*: *y q*

###

ln_{}

*q t*

###

*q t*

_{}for

*q*0. In correspondence with (2), the relationship between

*a*and

*y* is

*y* *a b q*. (9)
If *b*0 (no slope in Figure 3 bottom), we get *y**a* for any

*q*, i.e., *y* does not change with *q*; this is the case for the
(standard) exponential function.

III. THE INVERSE OF THE EXPONENTIAL INTEGRAL

FUNCTION Ei

###

*x*

With (7), one can calculate *t q*

###

, i.e., time as a function of quantity. To calculate*q t*

###

for given*t*, the inverse of the exponential integral function Ei

_{ }

*x*is needed. The inverse

( 1)

Ei _{}*z _{Ei}*

_{}

*x*, however, is more problematic to calculate than Ei

###

*x*

*z*. There are two reasons for this: First, for negative

_{Ei}*z*, Ei( 1)

_{Ei}_{}

*z*

_{Ei}_{}has always two solutions (see Figure 1 bottom), so that Ei( 1)

_{}

*z*

_{Ei}_{}does not represent a true function. One solution corresponds to the case where

*x*is positive in Figure 1 top, and the other where

*x*is negative. To address the problem of calculating the inverse, one has to distinguish between the two branches of the original function (Figure 1 top):

###

###

00

Ei for 0,

Ei for 0.

*x*
*Ei*

*x*

*x* *x*

*z*

*x* *x*

We define the solutions for the inverse in Figure 1 bottom
as two distinct (true) functions: *x z*

###

_{Ei}###

Ei( 1)**

_{x}_{}

_{0}

_{}

*z*

_{Ei}_{}and

###

*Ei*

###

*x z* Ei( 1)* _{x}*

_{}

_{0}

_{}

*z*

_{Ei}_{}. It is clear which case applies in (6):

*q*in Ei

_{}

*b q*

_{}

is always positive, and therefore
###

###

( 1) 0 ( 1) 0

with 0 Ei ,

Ei

with 0 Ei .

*Ei*

*Ei*
*x*

*Ei*
*x*

*b* *b q* *z*

*z* *b q*

*b* *b q* *z*

_{} _{ } _{} _{} _{}

_{} _{}

Figure 1. *Top*: The exponential integral function

###

1###

###

2###

###

Ei *x* ln*x**x* 1! 1 *x* 2! 2 ... with a singularity at

0

*x* , where both branches Ei_{x}_{}_{0}

###

*x*and Ei

_{x}_{}

_{0}

###

*x*approach

. Upwards, Ei_{x}_{}_{0}

###

*x*goes to , and Ei

_{x}_{}

_{0}

###

*x*to zero.

*Bottom*: For *z _{Ei}*0, the inverse of Ei

###

*x*

*z*presents two solutions of

_{Ei}*x*. Note that Ei( 1)

**

_{x}_{}

_{0}

###

###

0, corresponding to###

Ei 0 ; Ei( 1)*x* _{0}

###

*zEi*

###

0.372507

(when given with six

decimals); Ei( 1)* _{x}*

_{}

_{0}

###

0 ; and Ei( 1)*x*

_{0}

###

*zEi*

###

approaches quickly

zero for more negative *zEi*, so that one has to consider sufficient
accuracy of the numbers involved in the calculation of *zEi*.

The second reason, why the inverse function of Ei

_{ }

*x*represents a problem, is that there is no formula for calculating it. Only some functions for numerical approximations within certain intervals of

*x*exist (see [9]). One (not very elegant) way to calculate the inverse

2 1 0 1 2 3

5 0 5 10

*x*

E

i

*x*

*zEi*

Ei*x* 0*x*

Ei*x* 0*x*

5 0 5 10

2 1 0 1 2 3

*z _{Ei}*

*x*

Ei* _{x}* 0
1

*z _{Ei}*

Ei* _{x}* 0
1

*z _{Ei}*

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

indirectly is to repeatedly solve numerically Ei

_{ }

*x*

*z*: For example, Ei

_{Ei}###

1.5 is 3.301, when given with three decimals, so that the inverse of the exponential integral function at 3.301 is 1.5 (see Figure 1). To solve, for example,###

Ei *x* 3, one has to search iteratively below 1.5 to find *x*.
In the following, we present a new way of solving this
problem. We can express (4) as

###

1

Ei ln .

!
*k*
*k*

*x*

*x* *x*

*k k*

###

###

(10)The natural logarithm can be represented as an infinite series (page 137 in [6]):

2 1

1

2 1

ln for 0.

2 1 1

*k*

*k*

*x*

*x* *x*

*k* *x*

_{} _{}

_{} _{}

###

(11)Inserting (11) into (10) yields

###

2 1

1

1 2

Ei .

2 1 1 !

*k*

*k*

*k*

*x* *x*

*x*

*k* *x* *k k*

_{} _{} _{}

_{} _{} _{} _{}

###

It turns out that this is equivalent to

Ei

###

2 atanh 1 0, ln1
*x*

*x* *x* *x*

*x*

_{} _{} _{} _{}

, (12)

where atanh refers to the inverse hyperbolic tangent function (page 142 in [6]), and to the incomplete gamma function (page 236 in [6]):

1 1

atanh ln , and

2 1

*z*
*z*

*z*

1 _{.}

0, exp d

*x*

*x* *v* *v* *v*

_{}

With *z*

###

*x*1

###

*x*1

###

in atanh_{ }

*z*:

.

1 1 1 1

atanh 1 1

1 1

2 1

ln

2

*x* *x* *x*

*ln*

*x* *x*

*x*

*x*

_{} _{} _{} _{}

Substituting atanh_{}

###

*x*1

###

*x*1

###

_{}in (12), distinguishing between positive and negative

*x*, and simplifying, the following two equations are obtained:

###

###

0

0

Ei ln 1 0, ,

Ei 0, .

*x*

*x*

*x* *x*

*x* *x*

These equations provide a mathematical filter to distinguish between the two branches in Ei

_{ }

*x*, for

*x*0 and

*x*0, by keeping the branch of interest for given

*z*and either positive or negative

_{Ei}*x*, in the domain of real numbers, while moving the other branch into the domain of complex numbers. Not that in the case of

*x*0, ln

_{ }

1 *i* is

always in the domain of the complex logarithm, but the
imaginary part of Ei*x*0

###

*x*at the end becomes zero in the

solution.

Thus, the problem has been converted in the following
way: We wish to find *x* for given *z _{Ei}*, in order to find the
value of the inverse functions

*x z*

###

_{Ei}###

Ei*( 1)*

_{x}_{}

_{0}

_{}

*z*

_{Ei}_{}or

###

*Ei*

###

*x z* Ei( 1)* _{x}*

_{}

_{0}

_{}

*z*

_{Ei}_{}:

ln 1 0, for 0,

0, for 0.
*Ei*

*x* *x*

*z*

*x* *x*

(13)

Whereas this still does not provide an explicit formula for

###

*Ei*

###

*x z* , the advantage of (13) is that there is no ambiguity
anymore of two function values for Ei( 1)

_{ }

*x*, since the upper and lower branches have been separated (Figure 1 bottom). To find

*x*for given

*z*, one has to use a root-finding algorithm (see chapter 9 of [11]).

_{Ei}Concerning computation, the numerical evaluation of the
incomplete gamma function has been treated in [10], pages
221-224 of [11], and pages 560-567 of [12]. When using
*Mathematica*, the computation of *x z*

###

_{Ei}###

is summarized in Table 1. The formulas from (13) are given in Table 1, the core code is shown, as well as recommended starting values for the indirect search are provided (see also the section “Supporting Information” at the end of the article).TABLE 1. SUMMARY FOR CALCULATING NUMERICALLY THE
INVERSE OF THE EXPONENTIAL INTEGRAL FUNCTION *zEi*Ei

###

*x*

###

*Ei*

###

*x z* ( 1)

0

=Ei* _{x}*

_{}

_{}

*z*

_{Ei}_{}=Ei( 1)

**

_{x}_{}

_{0}

_{}

*z*

_{Ei}_{}

Formula

Given *z _{Ei}*, solve

*z*

_{Ei}###

###

###

ln 1 0, *x*

numerically for *x* in
the domain of real
numbers

Given *z _{Ei}*, with

0

*Ei*

*z* , solve

###

0,###

*Ei*

*z* *x*
numerically for *x* in
the domain of real
numbers (*z _{Ei}*0 does
not exist in this branch)

*Mathematica*

code

Re[ FindRoot[ zEi = =

Log[1] Gamma[0,

x], {x, zStart}, WorkingPrecision

>30, MaxIterations

>1000]

Re[ FindRoot[ zEi = =

Gamma[0, x], {x, zStart},

WorkingPrecision

>30, MaxIterations

>1000]

Recommended values for zStart

for 1: 10 *Ei*, and
*Ei*

*z* *z*

for *z _{Ei}*1: ln

_{}

*z*

_{Ei}_{}

for 0.5: exp , and

*Ei*

*Ei*

*z*
*z*

for 0.5 0:

ln

*Ei*

*Ei*

*z*
*z*

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

There is one closely related, as well as another, completely different approach to find numerically the inverse of the exponential integral function. First, one can use (5a) with ln

_{ }

*x*for Ei

_{x}_{}

_{0}

_{ }

*x*and ln

_{ }

*x*for

###

0Ei_{x}_{} *x* , instead of ln_{}*x*_{}. The accuracy is controlled
with the remainder of (5b). The second approach, given in
the supporting information, consists of a segment-wise
numerical simulation of the two branches of the function in
Figure 1 bottom, with high-degree polynomials (e.g., 31
coefficients) and a high number of calculated data points
(e.g., 10,000) in each of several segments.

IV. THE FINAL FORMULAS OF THE GENERALIZED

EXPONENTIAL FUNCTION

Employing the inverse of Ei

###

*x*, (6) can now be re-arranged to calculate

*q t*

###

, rather than*t q*

###

. With*t*

*tC*and

*qC*0, we define the generalized exponential function as:

###

###

###

###

###

###

###

( 1) 0

( 1) 0

exp 1

Ei for 0,

+Ei

exp exp for 0,

exp 1

Ei for 0,

+Ei

*C*
*C*

*C* *C*

*C*
*C*

*x*

*x*

*t t* *a*

*b*

*b* *b q*

*q t* *q* *t t* *a* * b*

*t t* *a*

*b*

*b* *b q*

_{ }_{ }

_{} _{} _{} _{}

_{} _{}

_{ }

_{} _{} _{} _{}

(14)

where *b*0 represents sigmoid growth, *b*0 exponential
growth, and *b*0 explosive growth, and where for *b*0
the restriction

###

*t t*

_{C}###

*exp a*

###

+*Ei*

###

*b q*

_{C}###

0 applies. Rearranging this restriction leads toEi for 0

exp

*C*
*C*

*b q*

*t* *t* *b*

*a*

. (15)

The restriction in (15) represents an upper bound for *t*, and
reflects the fact that for explosive growth with given
coefficients, there is a maximum time *t* that can be reached
for given parameters *a*, *b*, *qC*, and *tC*, no matter how large

*q* is.

We call (14) “generalized exponential function”. The
term has rarely been used before for other functions. In
unrelated form, the term “generalized (*q*-)exponential
function” is used in [13] (page 2923) to refer to

###

lim 1_{'}

###

'###

1 '*q*.

*q* _{q}_{q}

*e* *x* *q x* _{}

_{} _{} In [14] (page 574), the term

“generalized exponential growth model” is employed for

###

###

exp###

###

###

1*f t* *a* *b* *t* .

Note that in computations with (14) one can also use the
formulas for *b*0 arbitrarily close to *b*0, instead of
using the one for *b*0. One can show this with (6), the
formulas that form the basis for (14). Using (4) and *b*0,
the formulas convert into each other:

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

1 2

Ei 0 Ei 0

0 0

ln ln ...

1! 1 2! 2

ln ln .

*C*

*C* *C*

*C*

*C*

*q* *q*

*q q* *q q*

*q* *q*

*q* *q*

For sigmoid growth (*b*0) in (14), and *t* , *q t*

###

will not converge to a maximum quantity. Using (2), one can see that*q t*

###

*q t*

###

exp[*a b q t*

###

], which for increasing###

0*q t* remains always positive.

Equation (14) with *b*0 represents the formula for
calculating relative growth from a smaller *qC* to a larger

###

###

ln###

ln###

*C*

###

###

*C*

###

*q t* *q* *q* *t**t* , suggested already as the
“correct” formula by Sir Ronald Fisher in 1921 [15]. It
reflects exponential growth. Whereas this formula is
relatively well-known by field biologists, (14) with *b*0
for sigmoid growth and for explosive growth, as the more
general cases, are new.

For calculating the logarithmic relative growth at any
time, *q* in (9) is substituted by *q t*

###

from (14):

###

###

###

###

###

###

###

( 1) 0

( 1) 0

for 0 and 0,

for 0,

for 0 and 0,

exp Ei

+Ei

exp Ei

+Ei

*C*
*C*

*C*
*C*

*x*

*x*

*t*

*t*

* b* *a* *y*

* b*

* b* *a* *y*

*t t* *a*

*a*

*b q*

*y t* *a*

*t t* *a*

*a*

*b q*

_{ }

_{} _{} _{} _{}

_{} _{}

_{} _{}

_{} _{} _{}

(16)

where for *b*0 the restriction of (15) applies. With the
left-hand side *y t*

###

involved in the condition to choose among options for*b*0, this formula is only practical if one knows that

*a*

*y t*

###

will be positive or negative. Otherwise, if one does not want to calculate*t y*

###

instead of*y t*

###

, the alternative is to employ (9), where*q t*

###

has to be calculated first with (14).Furthermore, the instantaneous increment as a function of time is

###

###

###

###

###

exp for 0,

exp exp for 0,

*C* *C*

*t* *t*

*q* *a b q* * b*

*q t*

*q* *a* *t t* *a* *b*

_{} _{}

(17)

where in the case of *b*0, *q t*

###

has to be calculated first with (14). Introducing (14) for*q t*

###

in (17) would not lead to simplification, but would involve twice the inverse of###

Ei *x* , causing in turn complicated conditions if

( 1) 0

Ei* _{x}*

_{}

_{}

*z*

_{Ei}_{}or Ei( 1)

**

_{x}_{}

_{0}

_{}

*z*

_{Ei}_{}applies. For

*b*0, the formula is derived from (2). For

*b*0, the formula is derived from (14) by taking the derivative.

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

V. DETERMINATION OF THE FUNCTION PARAMETERS FROM THREE DATA POINTS

The generalized exponential function of (14) with *b*0 is
a three-parameter model, whose parameters *a*, *b*, and *q _{C}*
(at

*t*) or

_{C}*t*(at

_{C}*q*) are thus determined exactly by three points

_{C}*q*

_{1}at

*t*

_{1},

*q*

_{2}at

*t*

_{2}, and

*q*

_{3}at

*t*

_{3}, with

*t*

_{1}

*t*

_{2}

*t*

_{3}and

*q*

_{1}

*q*

_{2}

*q*

_{3}or

*q*

_{1}

*q*

_{2}

*q*

_{3}. Given such three data points, any one can represent

*q*at

_{C}*t*. To determine

_{C}*b*, one has to solve the following relationship numerically and indirectly (in

*Mathematica*with the FindRoot function, with

1

being generally a recommended starting value):

3 1 3 1

2 1

2 1

Ei Ei

Ei Ei

*b q* *b q* *t* *t*

*t* *t*

*b q* *b q*

_{} _{}

. (18)

The formula is derived by transforming the equation of (7)
for *b*0 with *q*_{1} at *t*_{1} being the calibrating point *q _{C}* at

*t*, resulting in exp

_{C}###

*a*

###

Ei###

*b q*

###

Ei *b q*1

###

###

*t*

###

*q*

*t*1

###

. By substituting*q*at

*t*

###

*q*with

*q*

_{2}at

*t*

_{2}on the one hand, and with

*q*

_{3}at

*t*

_{3}on the other, one obtains two equations for exp

###

*a*, which are set equal. Having determined

*b*, one calculates

3 1

3 1

Ei Ei

ln *b q* *b q* .

*a*

*t* *t*

_{} _{} _{} _{}

Since the generalized exponential function represents a
continuous spectrum of growth functions from sigmoid to
explosive, one expects to always find a solution for *b* in
(18), and subsequently for *a*. Note that the three points may
lie on a straight line, but that the resulting function will
represent nevertheless in that case a sigmoid curve among
the points.

An alternative method to determine *a* and *b* for *b*0 is
to use nonlinear regression with (6), which (at least in
*Mathematica*) works well. The regression equation is

###

###

###

###

###

1 Ei *b q* Ei *b q*1 exp *a*

*t**t* _{} _{} . With three data
points of *t*

###

*q*, the resulting coefficient of determination has to be 1. The regression can also be used for more than three data points, in which case it becomes a true regression analysis, with (generally) variance of the residuals and

2 _{1}

*R* . In that case, one can also let the regression
determine either

*q*

*C*or

*t*

*C*. Note, however, that the residuals present time, rather than quantity.

VI. INFLECTION POINT IN NON-EXPONENTIAL GROWTH

An inflection point refers to the point on a growth curve,
where the curve switches from being left-winged (locally
convex) to being right-winged (locally concave), or *vice *
*versa* (circles in Figure 2). Knowing and analyzing a

formula for the inflection point is of interest for several reasons:

(i) The inflection point characterizes the shape and scale
of a growth curve. In a nonlinear growth curve model,
there are few points on the curve that can be used to
compare different curves of a given function: In the
case of the generalized exponential function, there is
one calibrating point (*q _{C}* at

*t*), as well as for

_{C}*b*0 the inflection point (

*q*at

_{IP}*t*).

_{IP}(ii) The formula for calculating the location of the inflection point reveals important information about the dependence of the growth curve’s shape, location, and scale on the different parameters: How does changing the function’s parameters affect the inflection point, and consequently the whole growth curve?

(iii) It can also be of interest to develop a growth curve around a known inflection point, i.e., to derive the growth function’s parameters with a known (fixed) inflection point.

The inflection point quantity (*qIP*) is derived by taking
the second derivative of (6) for *b*0 with respect to *q*:

2 2

###

###

2. d

*t*

d*q* 1

*b q*exp

_{}

*a b q*

_{}

*q*(19) Setting this expression equal to zero, and solving for

*q*, yields:

*q _{IP}* 1

*b*, (20) which is positive only for

*b*0. This relation reveals a surprisingly simple interpretation of

*b*as the negative inverse of the inflection point quantity. For

*b*0, i.e., explosive growth,

*q*would be negative, which however has no real-world interpretation, and thus will not be considered further.

_{IP}Inserting (20) into (6) for *b*0 results in the
corresponding inflection point time (*tIP*):

Ei 1 Ei

for 0, exp

*C*

*IP* *C*

*b q*

*t* *t* *b*

*a*

(21)

where Ei 1

_{ }

1.89512, when given with five decimals.
Furthermore, as a necessary condition, one has to show
that there is a change of sign of the second-derivative at the
inflection point (page 231 in [16]). With *q*0, the second factor

_{exp}

###

_{ }

_{a}

_{b q}###

*2*

_{q}_{ in (19) will always be positive. }Therefore, if the second derivative will be positive or negative depends only on the first factor

###

1 *b q*

###

in (19). For*b*0,

###

1 *b q*

###

0 implies*q* 1

*b*, whereas

###

1*b q*

###

0 implies *q* 1*b*. For *b*0,

###

1*b q*

###

0 implies *q* 1*b*, whereas

###

1 *b q*

###

0 implies*q* 1 .

*b*

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

Figure 2. Variations of sigmoid growth curves (*b*0), when changing symmetrically the growth rate parameter *a*, the shape and scaling
parameter *b*, or the quantity *qC* of the calibrating point at *tC*0.5 in (14) with *b*0. *Top left*: Change of *a*0 (initial relative growth
is 100%) by 0.5. *Top right*: Change of *b* 0.2 (*qIP*5) by 0.1. The new inflection point quantity is *qIP*

###

1 *b b*

###

, and the new inflection point time has to be calculated with (21).*Bottom left*: Change of

*q*1 by 0.9 without changing

_{C}*b*, i.e., the inflection point quantity (circles) moves horizontally.

*Bottom right*: Changing

*qC*and

*b*according to (23) changes the growth curve’s scaling, and moves the inflection point quantity vertically. Note that relative growth remains the same for all three curves at any time in this graph, since

###

###

*y**a* *b f* *q f* *a* *b q*.

Consequently, there is always a change of sign at *qIP*.
Since the second derivative switches from negative to
positive at the inflection point with increasing *q*, the
formula also proves that the function switches always from
right-winged to left-winged for *b*0. Recall, however, that
(6) is the formula for *t*

###

*q*. In the case of

*q*

###

*t*in (14), the curve’s shape changes to its mirror image, resulting for

0

*b* always in a growth curve that switches inversely from
being left-winged to being right-winged.

Depending on *t _{C}*,

*b*, and

*q*, the inflection point time

_{C}*IP*

*t* can be positive or negative; a negative inflection point
time for *b*0 means that *t _{IP}*

*t*, i.e., only the

_{C}right-winged (concave) part of the growth curve appears for positive time.

VII. THE RELATIONSHIP AMONG THE PARAMETERS *a*,*b*,

AND *q _{C}* AT

*t*

_{C}In the generalized exponential function, the three parameters
*a*, *b*, and *q _{C}* at

*t*have straight-forward interpretations: Whereas

_{C}*q*at

_{C}*t*calibrates the position of the growth curve segment,

_{C}*a*is the initial logarithmic relative growth, i.e., the intercept at

*q*0 (see Figure 3 bottom). It serves as a standardized growth rate parameter (see Figure 1 in [8]).

2 4 6 8 10

0 5 10 15

Time*t*

Q

u

a

n

ti

ty

*q*

*b* 0.2

*a* 0.5

*a* 0

*a* 0.5

2 4 6 8 10

0 5 10 15 20 25

Time*t*

Q

u

a

n

ti

ty

*q*

*a* 0

*b* 0.3

*b* 0.2

*b* 0.1

2 4 6 8 10

0 5 10 15

Time*t*

Q

u

an

ti

ty

*q*

*a* 0,*b* 0.2

*qC* 0.1

*qC* 1

*qC* 1.9

2 4 6 8 10

0 5 10 15 20

Time*t*

Q

u

an

ti

ty

*q*

*a* 0

*b* 0.2 0.5

*b* 0.2

*b* 0.2 1.5

*qC* 1 0.5

*qC* 1

*qC* 1 1.5

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

On the other hand, *b* is a shape parameter in terms of its sign,
as well as a scaling parameter in terms of its magnitude.

Different combinations of parameter values between *a*
and *b* cause different curvatures. Note that the second
derivatives of (6) for *b*0, as measures of the
curvature, depend on both *a* and *b*:

d2_{2} Ei Ei ,

exp d

*C*

*b q* *b q*

*a*
*a*

*t*

_{}

_{}

_{}

_{}

(22a)

###

###

###

###

###

###

###

###

2

2 2

.

d 1

exp d

1 exp 1 exp

*C*

*C* *C*

*a b q* *q*

*b* *b*

*b q* *b q* *b q* *b q*

*t*

_{}

_{}

(22b)

For example, the more negative *a* in (22a), the larger is the
curvature. Extreme curvatures are associated with extreme
parameter values of *a* and *b*.

Figure 2 shows different combinations among the three
parameters *a*, *b*, and *q _{C}* in the case of sigmoid curves
(

*b*0). Changing

*t*is not shown in Figure 2, because it only moves the whole curve to the left or right, as can be seen from (6). The inflection points are indicated with circles in Figure 2. In Figure 2 top left, the parameter

_{C}*a*is varied around

*a*0 (i.e., initial relative growth is 100%). One sees that a symmetrical change of

*a*(here 0.5) leads to symmetrical growth curves around the original curve, with constant

*q*but different inflection point times

_{IP}*t*. This is not the case for a symmetrical change of

_{IP}*b*(here

0.1

), which results in asymmetrical curves around the
original growth curve (Figure 2 top right). In Figure 2
bottom left, *q _{C}* is moved up and down by the same amount
(here 0.9 ). The

*q*, however, are kept constant, which causes asymmetrical shapes of the resulting growth curves around the original one. Changing both the quantities of

_{IP}*q*and

_{C}*q*(via

_{IP}*b*) together by the same proportion results in a change of scale, shown in Figure 2 bottom right and explained next.

In addition to *b* determining if the growth curve is
sigmoid, exponential, or explosive, the variables *b* and *q _{C}*
are scaling

*q t*

###

. In (14) with*b*0, one can multiply

*q*by some factor, and divide

_{C}*b*by the same factor (say

*f*), to model a growth curve of

*qf*

###

*t*

*f q*

###

*t*:

###

###

###

###

###

( 1)

exp

Ei .

+Ei
*C*

*f*

*C*

*t t* *a*

*f*

*q* *t* _{b}

*b* _{q}_{f}

*f*

_{ }

_{} _{}

_{} _{}

(23)

The shape of the curve remains the same, with the new
inflection point quantity *f q* * _{IP}*, but unchanged inflection
point time. The factor

*f*does neither affect the initial logarithmic growth rate

*a*, nor

*y*in (9) or (16).

VIII. NEGATIVE GROWTH WITH THE GENERALIZED EXPONENTIAL FUNCTION

According to the standard exponential function, growth can
be positive or negative. To describe negative growth with
the generalized exponential function, presented in (14), one
has to replace *t t* * _{C}* with

*t*

_{C}*t*. We add the subindex

*NG*for negative growth, so that (14) changes to

###

###

###

###

###

###

###

( 1) 0

( 1) 0

exp 1

Ei for 0,

+Ei

exp exp for 0,

exp 1

Ei for 0,

+Ei

*C*

*C*

*NG* *C* *C*

*C*

*C*

*x*

*x*

*t* *t* *a*

*b*

*b* *b q*

*q* *t* *q* *t* *t* *a* * b*

*t* *t* *a*

*b*

*b* *b q*

_{ }

_{} _{} _{} _{}

_{} _{}

_{ }

_{} _{} _{} _{}

(24)

where *b*0 represents negative sigmoid growth, *b*0
negative exponential growth, and *b*0 negative explosive
growth. The restrictions in (8), when *b*0, convert to

for , and for .

*NG* *C* *C* *NG* *C* *C*

*q* *q* *t**t* *q* *q* *t**t*

Furthermore, (15) changes for negative growth:

for 0 and negative growth.

Ei

exp

*C*

*C* *b*

*b q*
*t* *t*

*a*

_{}

Finally, (21) converts to

for 0 & negative growth.

Ei 1 Ei

exp

*C*

*IP* *C* *b*

*b q*

*t* *t*

*a*

_{}

For negative growth, one also substitutes *t t* * _{C}* with

*t*

_{C}*t*in (7) for calculating

*a*, in (16) for calculating

*y t*

###

, and in (17) for calculating*q t*

###

.IX. GENERALIZED VERSUS STANDARD EXPONENTIAL

GROWTH

To show how the formulas work, and how the generalized
exponential function compares with the standard
exponential function, some sample growth curves are shown
in Figures 3 and 4. To get a sigmoid growth curve with
inflection point *t _{IP}*7,

*q*5, and calibrating point

_{IP}2
*C*

*t* , *q _{C}* 1, according to (20) one uses

*b* 0.2. To calculate the corresponding

*a*, one has to transform (21):

Ei 1 Ei

ln *C* for 0.

*IP* *C*

*b q*

*a* *b*

*t* *t*

_{ } _{} _{}

The restrictions from (8) become *q _{IP}*

*q*for

_{C}*t*

*t*, and

_{C}*IP* *C*

*q* *q* for *t**t _{C}*. For negative growth, one has to
substitute

*t*

_{C}*t*for

_{IP}*t*

_{IP}*t*, with

_{C}*q*

_{IP}*q*for

_{C}*t*

*t*, and

_{C}*q*

_{IP}*q*for

_{C}*t*

*t*. Here, one gets

_{C}*a* 0.610.

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

Figure 3. Sigmoid growth (*b*0), exponential growth (*b*0), and explosive growth (*b*0), positive (left) and negative (right). *Top*:
Quantity *q* as a function of time, as calculated with (14) for positive growth (left), and with *t t** _{C}* substituted by

*t*

_{C}*t*in (24) for negative growth (right). Whereas the parameter

*b*determines the shape and scale of the function,

*a*determines the relative growth rate, and the calibrating point

*qC*

###

*tC*the position of the growth curve in the upper graphs.

*Bottom*: The underlying linear relationship of logarithmic

relative growth *y* as a function of *q*, as calculated with (9) for positive or negative growth, which cannot be distinguished in these two
graphs. Whereas *b* is the slope in this functional relationship, *a* is the intercept at a quantity of zero.

Using (14) with *b*0, Figure 3 top left shows the
resulting sigmoid growth curve. Going through *q _{C}* 1 at

2
*C*

*t* , the curve reaches *q*16.2 at *t*25. Employing
(7) for *b*0, one can calculate the logarithmic relative
growth rate for an exponential growth curve that also goes
through *q*1 at *t*2 and *q*16.2 at *t*25, which
results in *a* 2.11. The corresponding growth curve,
calculated with (14), is shown in Figure 3 top left.

Negative growth is shown in Figure 3 top right,
employing (24) for negative sigmoid growth, negative
exponential growth, and negative explosive growth, with the
same parameters of *b* as for positive growth, but choosing

*IP*
*C*

*q* *q* (and calculating the corresponding parameters for
*a*).

Figure 3 bottom shows the corresponding linear
relationships between *q t*

###

and logarithmic relative growth, as given in (2). In those graphs,*a*is the intercept and

*b*is the slope of the lines (

*q*at

_{C}*t*does not appear in this mathematical relationship).

_{C}Figure 4 shows two more functional relationships for the
growth curves from Figure 3 top. For sigmoid growth, there
appears a maximum of the instantaneous increment with
time. This maximum is at the inflection point time.First, the
relationship between *t _{IP}* and

*q*is given bysubstituting

_{IP}*b* in (21) with *b* 1*q _{IP}* from (20), and rearranging:

Ei 1 exp Ei

exp

*C*

*IP* *C*

*IP*
*q*

*t* *t* *a*

*q*
*a*

_{} _{}

. (25)

0 5 10 15 20 25

0 5 10 15

Time*t*

Q

u

a

n

ti

ty

*q*

calibrating point 2, 1

inflection point 7, 5 sigmoid explosive

*t* 6.54

exponential

0 5 10 15 20 25

0 2 4 6 8 10

Time*t*

Q

u

a

n

ti

ty

*q*

calibrating point 2, 8

inflection point 7, 5

negative sigmoid

negative explosive

*t* 0.054

negative exponential

0 5 10 15

4 3 2 1 0

Quantity

L

o

g

a

ri

th

m

ic

re

la

ti

v

e

g

ro

w

th

*y*

*q*

sigmoid *b* 0.2
*a* 0.6100

*a*

0.6100

*a* 2.111

explosive

*b* 0.05

exponential *b* 0

0 2 4 6 8

3.0 2.5 2.0 1.5 1.0 0.5

Quantity*q*

L

o

g

a

ri

th

m

ic

re

la

ti

v

e

g

ro

w

th

*y*

*q*

sigmoid *b* 0.2
*a*

1.073

*a* 1.073

*a* 1.465

explosive

*b* 0.05

exponential *b* 0

**IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08**

Figure 4. Two additional functional relationships for the positive (left) and negative (right) growth curves of Figure 3 top. *Top*: Logarithmic
relative growth *y* as a function of time, calculated with (16) for positive growth, and with *t t**C* substituted by *tC**t* for negative growth.

*Bottom*: Instantaneous increment *q t*

###

as a function of time, calculated with (17) for positive growth, and with*t t*

*C*substituted by

*tC*

*t*for negative growth. For sigmoid growth (

*b*0), there is a maximum at the inflection point time, calculated with (26).

Next, to calculate the maximum instantaneous increment, one substitutes

*t*

in (17) for *b*0, with

*t*from (25), which yields

_{IP}

###

*1*

_{max}*exp a*1 for

*b*0.

*b*

*q t*

###

_{}

_{} (26)

The formula applies for both positive and negative growth.

X. CONVERSION OF A SEGMENTED CURVE OF *y t*

###

INTO AGROWTH CURVE OF *q t*

###

Assume that there are two known points of logarithmic
relative growth, *y _{i}* at

*t*and

_{i}*y*

_{i}_{}

_{1}at

*t*

_{i}_{}

_{1}, where the integer subindex

*i*serves to index and distinguish the points (a

larger *i* refers to a later time point). Then one can find the
corresponding *a _{i}* and

*b*for the generalized exponential function that goes from the first to the second point. The procedure will be useful, for example, for statistical applications. There may be more than two subsequent points

_{i}*i*

*y* at *t _{i}*, and one wants to use (16) to get a continuous
function through all points, employing splines (Figure 5 top
left). This segmented function can then be converted into a
growth curve of quantity as a function of time (Figure 5
bottom left). The method works only for segments with

0,

*b* but *b* can be arbitrarily close to 0.

Transforming (16) for *b*0, one gets two equations for
a single segment of the generalized exponential function,
where the subindex *i* refers to the function segment (spline)
in between time points *t _{i}* and

*t*

_{i}_{}

_{1}:

0 5 10 15 20 25

4 3 2 1 0 1 2

Time*t*

L

o

g

a

ri

th

m

ic

re

la

ti

v

e

g

ro

w

th

*y*

*t*

sigmoid explosive

exponential

0 5 10 15 20 25

3 2 1 0 1 2

Time*t*

L

o

g

a

rit

h

m

ic

re

la

ti

v

e

g

ro

w

th

*y*

*t*

negative sigmoid negative explosive

negative exponential

0 5 10 15 20 25

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Time *t*

In

c

re

m

e

n

t

*q*

'

*t*

sigmoid maximum 7, 0.999

explosive

exponential

0 5 10 15 20 25

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Time*t*

In

c

re

m

e

n

t

*q*

'

*t*

negative sigmoid maximum 7, 0.629

negative explosive

negative exponential