§ 1. Derivatives of the exponential and circular functions

1. Derivatives of the exponential functions; the number $e$

We know that every continuous homomorphism of the additive group $\mathbf{R}$ into the multiplicative group $\mathbf{R}^*$ of real numbers $\neq 0$ is a function of the form $x \mapsto a^x$ (called an exponential function) where $a$ is a number $> 0$ (TG, V, p.11); it is an isomorphism of $\mathbf{R}$ onto the multiplicative group $\mathbf{R}_+^*$ of numbers $> 0$ if $a \neq 1$, and the inverse isomorphism from $\mathbf{R}_+^*$ onto $\mathbf{R}$ is denoted by $\log_a x$ and is called the logarithm to the base $a$.

We shall see that the function $f(x) = a^x$ has, for every $x \in \mathbf{R}$, a derivative of the form $c.a^x$ (where clearly $c = f'(0)$). This results from the following general theorem:

We usually omit the base in the notation for the Naperian logarithm. Unless it is stated to the contrary, the notation $\log x$ ($x > 0$) will denote the Naperian logarithm of $x$. With this notation, prop. 1 can be written as the identity

valid for any $a > 0$ ($\log a = 0$ when $a = 1$).

This relation shows that $a^x$ has derivatives of every order, and that

In particular, for $a > 0$ and $\neq 1$ one has $D^2(a^x) > 0$ for all $x \in \mathbf{R}$, and hence $a^x$ is strictly convex on $\mathbf{R}$ (I, p. 31, corollary). From this one deduces the following proposition:

2. Derivative of $\log_a x$

Since $a^x$ is strictly monotone on $\mathbf{R}$ for $a \neq 1$, applying the rule for differentiating inverse functions (I, p. 17, prop. 6) gives, for all $x > 0$

If $u$ is a real function admitting a derivative at the point $x_0$ and such that $u(x_0) > 0$, then the function $\log u$ admits a derivative equal to $u'(x_0)/u(x_0)$ at the point $x_0$. In particular, we have $D(\log |x|) = 1/|x| = 1/x$ if $x > 0$, and

3. Derivatives of the circular functions; the number $\pi$

We have defined, in General Topology (Gen. Top., VIII, p. 106), the continuous homomorphism $x \mapsto \mathbf{e}(x)$ of the additive group $\mathbf{R}$ onto the multiplicative group $\mathbf{U}$ of complex numbers of absolute value 1; this is a periodic function with principal period 1, and $\mathbf{e}\left(\frac{1}{4}\right) = i$. One knows (loc. cit.) that every continuous homomorphism of $\mathbf{R}$ onto $\mathbf{U}$ is of the form $x \mapsto \mathbf{e}(x/a)$, and one puts $\cos_a x = \mathscr{R}(\mathbf{e}(x/a)), \sin_a x = \mathscr{I}(\mathbf{e}(x/a))$ (trigonometric functions, or circular functions, to base $a$); these last functions are continuous maps from $\mathbf{R}$ into $[-1, +1]$ having principal period $a$. We have $\sin_a(x + a/4) = \cos_a x, \quad \cos_a(x + a/4) = -\sin_a x$, and the function $\sin_a x$ is increasing on the interval $[-a/4, a/4]$.

4. Inverse circular functions

The restriction of the function $\sin x$ to the interval $[-\pi/2, +\pi/2]$ is strictly increasing; one denotes its inverse by $\operatorname{Arc}\sin x$, which is thus a strictly increasing continuous map of the interval $[-1, +1]$ onto $[-\pi/2, +\pi/2]$ (fig. 6). The formula for differentiating inverse functions (I, p. 9, prop. 6) gives the derivative of this function

Since $-\pi/2 \leq \operatorname{Arc}\sin x \leq \pi/2$ we have $\cos(\operatorname{Arc}\sin x) \geq 0$, and since

!Graph showing y = Arc cos x and y = Arc sin x

Fig. 6

Finally, the restriction of tan x to the interval ]−π/2, +π/2[ is strictly increasing; one denotes its inverse by Arc tan x, and this is a strictly increasing map from \mathbf{R} onto ]−π/2, +π/2[ (fig. 7); we have

and, by applying the formula for differentiating inverse functions and formula (15) of III, p. 95, we have

!Graph showing y = Arc tan x

Fig. 7

5. The complex exponential

We have determined (Gen. Top., VIII, p. 106) all the continuous homomorphisms of the (additive) topological group \mathbf{C} of complex numbers onto the (multiplicative) topological group \mathbf{C}^* of complex numbers ≠ 0; these are the maps

where $\alpha, \beta, \gamma, \delta$ are four real numbers subject to the single condition $\alpha \delta - \beta \gamma \neq 0$. We now propose to determine which of these homomorphisms $z \mapsto f(z)$ are differentiable on \mathbf{C}. First we remark that it is enough for $f$ to be differentiable at the point $z = 0$; indeed, for every point $z \in \mathbf{C}$ one has

if $f'(0)$ exists, then so does $f'(z)$, and $f'(z) = a f(z)$, with $a = f'(0)$. On the other hand, if $g$ is a second differentiable homomorphism, such that $g'(z) = b g(z)$, then $g(a z / b) = f(z)$, for one notes immediately that the quotient $g(a z / b) / f(z)$ has an everywhere zero derivative and is equal to 1 for $z = 0$; all the differentiable homomorphisms are thus of the form $z \mapsto f(\lambda z)$, where $f$ is one of them (assuming they exist) and $\lambda$ is any (complex) constant.

This being so, if $f$ is differentiable at the point $z = 0$ then each of the maps $x \mapsto f(x),\ y \mapsto f(iy)$ of $\mathbf{R}$ into $\mathbf{C}$ is necessarily differentiable at the point $0$, the first having derivative $f'(0)$, the second $if'(0)$. Now the derivatives of the maps $x \mapsto e^{\alpha x} \mathbf{e}(\gamma x),\ y \mapsto e^{\beta y} \mathbf{e}(\delta y)$ at the point $0$ are respectively equal to $\alpha + 2\pi i \gamma$ and $\beta + 2\pi i \delta$, from which $\beta = -2\pi \gamma$ and $\alpha = 2\pi \delta$; these conditions are, in particular, satisfied by the homomorphism $x + iy \mapsto e^z \mathbf{e}(y/2\pi)$, which we shall denote provisionally by $f_0$. We shall now show that $f_0$ is actually differentiable at the point $z = 0$.

It is clear that $x \mapsto f_0(x)$ and $y \mapsto f_0(iy)$ have derivatives of every order; in particular, Taylor's formula of order 1 applied to these functions shows that for every $\varepsilon > 0$ there is an $r > 0$ such that, if one puts

then the conditions $|x| \leq r,\ |y| \leq r$ imply that $|\varphi(x)| \leq \varepsilon$ and $|\psi(y)| \leq \varepsilon$; this being so, we have $f_0(x + iy) = f_0(x)f_0(iy) = 1 + (x + iy) + \theta(x,\ y)$ with

for $|z| \leq r$ we have $|x| \leq r$ and $|y| \leq r$, whence

which proves that the quotient $\frac{f_0(z) - 1 - z}{z}$ tends to 0 with $z$, that is to say, the function $f_0$ admits a derivative equal to 1 at the point $z = 0$. The above thus proves that, for all $z \in \mathbf{C}$,

This property establishes the connection between $f_0$ and the function $e^z$, which is moreover the restriction of $f_0$ to the real axis; for this reason we make the following definition:

Definition 2 The homomorphism $x + iy \mapsto e^z \mathbf{e}(y/2\pi)$ of $\mathbf{C}$ onto $\mathbf{C}^*$ is called the complex exponential; its value at an arbitrary complex number $z$ is denoted by $e^z$ or $\exp z$.

6. Properties of the function $e^z$

The fact that $z \mapsto e^z$ is a homomorphism of $\mathbf{C}$ onto $\mathbf{C}^*$ may be expressed by the identities

By definition, one has, for every $z = x + iy$,

and since $e^{\lambda} > 0$ one sees that $e^z$ has absolute value $e^{\lambda}$ and amplitude $y$ (modulo $2\pi$).

In particular, def. 2 (III, p. 98) gives

Since $2\pi$ is the principal period of $\mathbf{e}(y/2\pi)$, $2\pi i$ is the principal period of $e^z$; in other words, the group of periods of $e^z$ is the set of numbers $2n\pi i$, where $n$ runs through $\mathbf{Z}$.

Finally, formula (21) of III, p. 98 can be written

7. The complex logarithm

Let B be the "strip" formed by the points $z = x + i y$ such that $-\pi \leq y < \pi$; the function $e^z$ takes each of its values once and only once on B; in other words, $z \mapsto e^z$ is a bijective continuous map of B onto $\mathbf{C}^*$; the image under this map of the (half-open) segment $x = x_0,\ -\pi \leq y < \pi$ is the circle $|z| = e^{x_0}$; the image of the line $y = y_0$ is the (open) half-line defined by $\mathrm{Am}(z) = y_0$ (mod. $2\pi$). The image under $z \mapsto e^z$ of the interior $\dot{B}$ of B, that is, of the set of $z \in \mathbf{C}$ such that $|\mathscr{I}(z)| < \pi$, is the complement F of the (closed) negative real half-axis in $\mathbf{C}$; if one agrees to denote by $\mathrm{Am}(z)$ the measure of the amplitude of $z$ which belongs to $[-\pi, \pi[$, then the set F can be defined by the relations $-\pi < \mathrm{Am}(z) < \pi$. Since $z \mapsto e^z$ is a strict homomorphism of $\mathbf{C}$ onto $\mathbf{C}^*$ the image under this map of any open subset of B (so of $\mathbf{C}$) is an open set in $\mathbf{C}^*$ (so in F); in other words, the restriction of $z \mapsto e^z$ to B is a homeomorphism of $\dot{B}$ onto F. We denote by $z \mapsto \log z$ the homeomorphism of F onto B which is the inverse of the latter; for a complex number $z \in F$, $\log z$ is called the principal value of the logarithm of $z$. If $z = x + i y$ and $\log z = u + i v$ then $x + i y = e^{u+i v}$, whence $e^u = |z|$, and since $-\pi < v < \pi$, we have $v = \mathrm{Am}(z)$. Moreover, we have $\tan(v + \pi/2) = -x/y$ if $y \neq 0$; thus we can write

It is clear that $\log z$ is the extension to F of the function $\log x$ defined on the positive open real half-axis $\mathbf{R}_+^*$. If $z,\ z'$ are two points of F such that $zz'$ is not real negative, we have $\log(zz') = \log z + \log z' + 2\varepsilon \pi i$, where $\varepsilon = +1,\ -1$ or 0 depending on the values of $\mathrm{Am}(z)$ and $\mathrm{Am}(z')$.

We note that at the points of the negative real half-axis the function $\log z$ has no limit; to be precise, if $x$ tends to $x_0 < 0$ and if $y$ tends to 0 remaining > 0 (resp. < 0), then $\log z$ tends to $\log |x_0| + \pi i$ (resp. $\log |x_0| - \pi i$); when $z$ tends to 0, $|\log z|$ increases indefinitely.

We shall see later how the theory of analytic functions allows us to extend the function $\log z$, and to define the complex logarithm in full generality.

Since $\log z$ is the inverse homeomorphism of $e^z$, the formula for differentiating inverse functions (I, p. 9, prop. 6) shows that $\log z$ is differentiable at every point $z \in \mathbf{F}$, and that

8. Primitives of rational functions

Formula (31) allows us to evaluate the primitive of an arbitrary rational function $r(x)$ of a real variable $x$, with real or complex coefficients. We know (A.VII.7) that such a function can be written (in unique manner) as a finite sum of terms, which are:

a) either of the form $ax^p$ ($p$ an integer $\geqslant 0$, $a$ a complex number); b) or of the form $a/(x-b)^m$ ($m$ an integer $\geqslant 0$, $a$ and $b$ complex numbers).

Now it is easy to obtain a primitive of each of these terms: a) a primitive of $ax^p$ is $a \frac{x^{p+1}}{p+1}$; b) if $m > 1$ a primitive of $a/(x-b)^m$ is $\frac{a}{(1-m)(x-b)^{m-1}}$; c) finally, from formulae (10) (III, p. 93) and (31) (III, p. 101), a primitive of $\frac{a}{x-b}$ is $a \log |x-b|$ if $b$ is real, $a \log(x-b)$ if $b$ is complex. In the last case, if $b = p + iq$, one has furthermore (III, p. 100, formulae (30))

We postpone the examination of more practical methods for explicitly determining a primitive of a rational function given explicitly to the part of this work dedicated to Numerical Calculus.

One can reduce to the evaluation of a primitive of a rational function:

1° the evaluation of a primitive of a function of the form $r(e^{a x})$, where $r$ is a rational function and $a$ a real number; indeed by the change of variable $u = e^{a x}$ one reduces to finding a primitive for $r(u)/u$;

2° the evaluation of a primitive of a function of the form $f(\sin a x, \cos a x)$, where $f$ is a rational function of two variables and $a$ is a real number; the change of variables $u = \tan(a x/2)$ reduces this to finding a primitive for

9. Complex circular functions; hyperbolic functions

Euler’s formulae (25) (III, p. 99) and the definition of $e^z$ for every complex $z$ allow us to extend to $\mathbf{C}$ the functions $\cos x$ and $\sin x$ defined on $\mathbf{R}$, by putting, for all $z \in \mathbf{C}$

These functions are periodic with principal period $2 \pi$; that is, one has $\cos(z + \pi/2) = -\sin z, \sin(z + \pi/2) = \cos z$; one may also verify the identities

More generally, every algebraic identity between circular functions for real variables remains true when one gives these variables arbitrary complex values (III, p. 119, exerc. 18).

One puts $\tan z = \sin z / \cos z$ if $z \neq (2k + 1)\pi/2$ and $\cot z = \cos z / \sin z$ if $z \neq k \pi$; these are periodic functions with principal period $\pi$.

Formula (27) (III, p. 99) shows that $\cos z$ and $\sin z$ are differentiable on $\mathbf{C}$, and that

For $z = i x$ ($x$ real), the formulae (32) give

It is convenient to have a specific notation for the real functions thus introduced; we put

One thus has, for every real $x$

From every identity between circular functions in a certain number of complex numbers $z_k$ ($1 \leq k \leq n$) one can deduce an identity for the hyperbolic functions, by replacing $z_k$ by $i x_k$ ($x_k$ real, $1 \leq k \leq n$) and using the formulae (34); for instance one has

The hyperbolic functions allow us to express the real and imaginary parts of $\cos z$ and $\sin z$ for $z = x + i y$, since