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''exponent''
 
''exponent''
  
 
The function
 
The function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369101.png" /></td> </tr></table>
+
$$
 +
= e ^ {z}  \equiv  \mathop{\rm exp}  z ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369102.png" /> is the base of the natural logarithm, which is also known as the Napier number. This function is defined for any value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369103.png" /> (real or complex) by
+
where e $
 +
is the base of the natural logarithm, which is also known as the Napier number. This function is defined for any value of $  z $(
 +
real or complex) by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
e ^ {z}  = \lim\limits _ {n \rightarrow \infty }  \left ( 1 +
 +
\frac{z }{n}
 +
\right ) ^ {n} ,
 +
$$
  
 
and has the following properties:
 
and has the following properties:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369105.png" /></td> </tr></table>
+
$$
 +
e ^ {z _ {1} } e ^ {z _ {2} }  = \
 +
e ^ {z _ {1} + z _ {2} } \ \
 +
\textrm{ and } \ \
 +
( e ^ {z _ {1} } ) ^ {z _ {2} }  = \
 +
e ^ {z _ {1} z _ {2} }
 +
$$
  
for any values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369107.png" />.
+
for any values of $  z _ {1} $
 +
and $  z _ {2} $.
  
For real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369108.png" />, the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e0369109.png" /> (the exponential curve) passes through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691010.png" /> and tends asymptotically to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691011.png" />-axis (see Fig.).
+
For real $  x $,  
 +
the graph of $  y = e ^ {x} $(
 +
the exponential curve) passes through the point $  ( 0, 1) $
 +
and tends asymptotically to the $  x $-
 +
axis (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036910a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e036910a.gif" />
Line 21: Line 52:
 
Figure: e036910a
 
Figure: e036910a
  
In mathematical analysis one considers the exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691012.png" /> for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691015.png" />; this function is related to the (basic) exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691016.png" /> by
+
In mathematical analysis one considers the exponential function $  y = a  ^ {x} $
 +
for real $  x $
 +
and  $  a > 0 $,  
 +
$  a \neq 1 $;  
 +
this function is related to the (basic) exponential function $  y = e ^ {x} $
 +
by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691017.png" /></td> </tr></table>
+
$$
 +
a  ^ {x}  = e ^ {x  \mathop{\rm ln}  a } .
 +
$$
  
The exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691018.png" /> is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691019.png" /> and is positive, monotone (it increases if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691020.png" /> and decreases if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691021.png" />), continuous, and infinitely differentiable; moreover,
+
The exponential function $  y = a  ^ {x} $
 +
is defined for all $  x $
 +
and is positive, monotone (it increases if $  a > 1 $
 +
and decreases if $  0 < a < 1 $),  
 +
continuous, and infinitely differentiable; moreover,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691022.png" /></td> </tr></table>
+
$$
 +
( a  ^ {x} )  ^  \prime  = a  ^ {x}  \mathop{\rm ln}  a ,\ \
 +
\int\limits a  ^ {x}  dx  =
 +
\frac{a  ^ {x} }{ \mathop{\rm ln}  a }
 +
+ C,
 +
$$
  
 
and in particular
 
and in particular
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691023.png" /></td> </tr></table>
+
$$
 +
( e  ^ {x} )  ^  \prime  = e  ^ {x} ,\ \
 +
\int\limits e  ^ {x}  dx  = e ^ {x} + C ,
 +
$$
  
 
and in a neighbourhood of each point the exponential function can be expanded in a power series, for example:
 
and in a neighbourhood of each point the exponential function can be expanded in a power series, for example:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
e  ^ {x}  = \
 +
1 +
 +
\frac{x}{1!}
 +
+ \dots +
 +
\frac{x  ^ {n} }{n!}
 +
+ \dots \equiv \
 +
\sum _ { n= } 0 ^  \infty 
 +
\frac{x  ^ {n} }{n!}
 +
.
 +
$$
  
The graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691025.png" /> is symmetric about the ordinate axis to the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691028.png" /> increases more rapidly than any power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691029.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691030.png" />, while as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691031.png" /> it tends to zero more rapidly than any power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691032.png" />, i.e. for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691033.png" />,
+
The graph of $  y = a  ^ {x} $
 +
is symmetric about the ordinate axis to the graph of $  y = ( 1/a)  ^ {x} $.  
 +
If $  a > 1 $,  
 +
$  a  ^ {x} $
 +
increases more rapidly than any power of $  x $
 +
as $  x \rightarrow + \infty $,  
 +
while as $  x \rightarrow - \infty $
 +
it tends to zero more rapidly than any power of $  1/x $,  
 +
i.e. for any natural number $  b > 0 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691034.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow + \infty } 
 +
\frac{a  ^ {x} }{| x |  ^ {b} }
 +
  = \infty ,\ \
 +
\lim\limits _ {x \rightarrow - \infty }  | x |  ^ {b} a  ^ {x}  = 0.
 +
$$
  
 
The inverse of an exponential function is a [[Logarithmic function|logarithmic function]].
 
The inverse of an exponential function is a [[Logarithmic function|logarithmic function]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691036.png" /> are complex, the exponential function is related to the (basic) exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691037.png" /> by
+
If $  a $
 +
and $  z $
 +
are complex, the exponential function is related to the (basic) exponential function $  w = e ^ {z} $
 +
by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691038.png" /></td> </tr></table>
+
$$
 +
a  ^ {z}  = e ^ {z  \mathop{\rm Ln}  a } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691039.png" /> is the logarithm of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691040.png" />.
+
where $  \mathop{\rm Ln}  a $
 +
is the logarithm of the complex number $  a $.
  
The exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691041.png" /> is a transcendental function and is the analytic continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691042.png" /> from the real axis into the complex plane.
+
The exponential function $  w = e ^ {z} $
 +
is a transcendental function and is the analytic continuation of $  y = e ^ {x} $
 +
from the real axis into the complex plane.
  
 
An exponential function can be defined not only by (1) but also by means of the series (2), which converges throughout the complex plane, or by Euler's formula
 
An exponential function can be defined not only by (1) but also by means of the series (2), which converges throughout the complex plane, or by Euler's formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691043.png" /></td> </tr></table>
+
$$
 
+
e  ^ {z}  = e ^ {x+ iy }  = e ^ {x} ( \cos  y + i  \sin  y ).
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691044.png" />, then
+
$$
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691045.png" /></td> </tr></table>
 
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691046.png" /> is periodic with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691047.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691048.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691049.png" /> assumes all complex values except zero; the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691050.png" /> has an infinite number of solutions for any complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691051.png" />. These solutions are given by
+
If  $  z = x + iy $,
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691052.png" /></td> </tr></table>
+
$$
 +
| e  ^ {z} |  = e  ^ {x} ,\ \
 +
\mathop{\rm Arg}  e  ^ {z}  = y + 2 \pi k,\ \
 +
k = 0, \pm  1, \pm  2 , .  . . .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691053.png" /> is one of the basic elementary functions. It is used to express, for example, the trigonometric and hyperbolic functions.
+
The function $  e  ^ {z} $
 +
is periodic with period  $  2 \pi i $:
 +
$  e ^ {z + 2 \pi i } = e  ^ {z} $.  
 +
The function  $  e  ^ {z} $
 +
assumes all complex values except zero; the equation  $  e ^ {z} = a $
 +
has an infinite number of solutions for any complex number  $  a \neq 0 $.  
 +
These solutions are given by
  
 +
$$
 +
z  =  \mathop{\rm Ln}  a  =  \mathop{\rm ln}  | a | + i  \mathop{\rm Arg}  a.
 +
$$
  
 +
The function  $  e  ^ {z} $
 +
is one of the basic elementary functions. It is used to express, for example, the trigonometric and hyperbolic functions.
  
 
====Comments====
 
====Comments====
 
For Euler's formula see also [[Euler formulas|Euler formulas]].
 
For Euler's formula see also [[Euler formulas|Euler formulas]].
  
The basic exponential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691054.png" /> defined by (1) or, equivalently, (2) (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691055.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691056.png" />) is single-valued. However, powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691057.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691058.png" /> complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691059.png" /> are multiple-valued since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691060.png" /> denotes the  "multiple-valued inverse"  to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691061.png" />. Thus, since it is customary to abbreviate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691062.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691063.png" />, the left-hand side of the identity
+
The basic exponential function $  z \mapsto  \mathop{\rm exp} ( z) $
 +
defined by (1) or, equivalently, (2) (with $  z $
 +
instead of $  x $)  
 +
is single-valued. However, powers $  z \mapsto a  ^ {z} $
 +
for $  a $
 +
complex $  ( a \neq 0) $
 +
are multiple-valued since $  z \mapsto  \mathop{\rm Ln}  z $
 +
denotes the  "multiple-valued inverse"  to $  z \mapsto  \mathop{\rm exp} ( z) $.  
 +
Thus, since it is customary to abbreviate $  \mathop{\rm exp} ( z) $
 +
as e ^ {z} $,  
 +
the left-hand side of the identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691064.png" /></td> </tr></table>
+
$$
 +
( e ^ {z _ {1} } ) ^ {z _ {2} }  = \
 +
e ^ {z _ {1} z _ {2} }
 +
$$
  
 
is multiple-valued, while the right-hand side is single-valued. This identity is a dangerous one and should be dealt with with care, otherwise it may lead to nonsense like
 
is multiple-valued, while the right-hand side is single-valued. This identity is a dangerous one and should be dealt with with care, otherwise it may lead to nonsense like
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691065.png" /></td> </tr></table>
+
$$
 +
= 1  ^ {1/2}  = \
 +
( e ^ {2 \pi i } )  ^ {1/2}  = \
 +
e ^ {\pi i }  = - 1.
 +
$$
  
By considering a single-valued branch of the logarithm (cf. [[Branch of an analytic function|Branch of an analytic function]]), or by considering the [[Complete analytic function|complete analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691066.png" /> on its associated Riemann surface, an awkward notation and a lot of confusion may disappear. For fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691067.png" />, any value (i.e. determination) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691068.png" /> defines an exponential function:
+
By considering a single-valued branch of the logarithm (cf. [[Branch of an analytic function|Branch of an analytic function]]), or by considering the [[Complete analytic function|complete analytic function]] $  \mathop{\rm Ln} $
 +
on its associated Riemann surface, an awkward notation and a lot of confusion may disappear. For fixed $  a \in \mathbf C \setminus  0 $,  
 +
any value (i.e. determination) of $  \mathop{\rm Ln}  a $
 +
defines an exponential function:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036910/e03691069.png" /></td> </tr></table>
+
$$
 +
a  ^ {z}  = \
 +
e ^ {z ( \textrm{ value }  \textrm{ of }  \mathop{\rm Ln}  a) } .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Dieudonné,  "Foundations of modern analysis" , '''1''' , Acad. Press  (1969)  pp. 192  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Dieudonné,  "Foundations of modern analysis" , '''1''' , Acad. Press  (1969)  pp. 192  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR></table>

Revision as of 19:38, 5 June 2020


exponent

The function

$$ y = e ^ {z} \equiv \mathop{\rm exp} z , $$

where $ e $ is the base of the natural logarithm, which is also known as the Napier number. This function is defined for any value of $ z $( real or complex) by

$$ \tag{1 } e ^ {z} = \lim\limits _ {n \rightarrow \infty } \left ( 1 + \frac{z }{n} \right ) ^ {n} , $$

and has the following properties:

$$ e ^ {z _ {1} } e ^ {z _ {2} } = \ e ^ {z _ {1} + z _ {2} } \ \ \textrm{ and } \ \ ( e ^ {z _ {1} } ) ^ {z _ {2} } = \ e ^ {z _ {1} z _ {2} } $$

for any values of $ z _ {1} $ and $ z _ {2} $.

For real $ x $, the graph of $ y = e ^ {x} $( the exponential curve) passes through the point $ ( 0, 1) $ and tends asymptotically to the $ x $- axis (see Fig.).

Figure: e036910a

In mathematical analysis one considers the exponential function $ y = a ^ {x} $ for real $ x $ and $ a > 0 $, $ a \neq 1 $; this function is related to the (basic) exponential function $ y = e ^ {x} $ by

$$ a ^ {x} = e ^ {x \mathop{\rm ln} a } . $$

The exponential function $ y = a ^ {x} $ is defined for all $ x $ and is positive, monotone (it increases if $ a > 1 $ and decreases if $ 0 < a < 1 $), continuous, and infinitely differentiable; moreover,

$$ ( a ^ {x} ) ^ \prime = a ^ {x} \mathop{\rm ln} a ,\ \ \int\limits a ^ {x} dx = \frac{a ^ {x} }{ \mathop{\rm ln} a } + C, $$

and in particular

$$ ( e ^ {x} ) ^ \prime = e ^ {x} ,\ \ \int\limits e ^ {x} dx = e ^ {x} + C , $$

and in a neighbourhood of each point the exponential function can be expanded in a power series, for example:

$$ \tag{2 } e ^ {x} = \ 1 + \frac{x}{1!} + \dots + \frac{x ^ {n} }{n!} + \dots \equiv \ \sum _ { n= } 0 ^ \infty \frac{x ^ {n} }{n!} . $$

The graph of $ y = a ^ {x} $ is symmetric about the ordinate axis to the graph of $ y = ( 1/a) ^ {x} $. If $ a > 1 $, $ a ^ {x} $ increases more rapidly than any power of $ x $ as $ x \rightarrow + \infty $, while as $ x \rightarrow - \infty $ it tends to zero more rapidly than any power of $ 1/x $, i.e. for any natural number $ b > 0 $,

$$ \lim\limits _ {x \rightarrow + \infty } \frac{a ^ {x} }{| x | ^ {b} } = \infty ,\ \ \lim\limits _ {x \rightarrow - \infty } | x | ^ {b} a ^ {x} = 0. $$

The inverse of an exponential function is a logarithmic function.

If $ a $ and $ z $ are complex, the exponential function is related to the (basic) exponential function $ w = e ^ {z} $ by

$$ a ^ {z} = e ^ {z \mathop{\rm Ln} a } , $$

where $ \mathop{\rm Ln} a $ is the logarithm of the complex number $ a $.

The exponential function $ w = e ^ {z} $ is a transcendental function and is the analytic continuation of $ y = e ^ {x} $ from the real axis into the complex plane.

An exponential function can be defined not only by (1) but also by means of the series (2), which converges throughout the complex plane, or by Euler's formula

$$ e ^ {z} = e ^ {x+ iy } = e ^ {x} ( \cos y + i \sin y ). $$

If $ z = x + iy $, then

$$ | e ^ {z} | = e ^ {x} ,\ \ \mathop{\rm Arg} e ^ {z} = y + 2 \pi k,\ \ k = 0, \pm 1, \pm 2 , . . . . $$

The function $ e ^ {z} $ is periodic with period $ 2 \pi i $: $ e ^ {z + 2 \pi i } = e ^ {z} $. The function $ e ^ {z} $ assumes all complex values except zero; the equation $ e ^ {z} = a $ has an infinite number of solutions for any complex number $ a \neq 0 $. These solutions are given by

$$ z = \mathop{\rm Ln} a = \mathop{\rm ln} | a | + i \mathop{\rm Arg} a. $$

The function $ e ^ {z} $ is one of the basic elementary functions. It is used to express, for example, the trigonometric and hyperbolic functions.

Comments

For Euler's formula see also Euler formulas.

The basic exponential function $ z \mapsto \mathop{\rm exp} ( z) $ defined by (1) or, equivalently, (2) (with $ z $ instead of $ x $) is single-valued. However, powers $ z \mapsto a ^ {z} $ for $ a $ complex $ ( a \neq 0) $ are multiple-valued since $ z \mapsto \mathop{\rm Ln} z $ denotes the "multiple-valued inverse" to $ z \mapsto \mathop{\rm exp} ( z) $. Thus, since it is customary to abbreviate $ \mathop{\rm exp} ( z) $ as $ e ^ {z} $, the left-hand side of the identity

$$ ( e ^ {z _ {1} } ) ^ {z _ {2} } = \ e ^ {z _ {1} z _ {2} } $$

is multiple-valued, while the right-hand side is single-valued. This identity is a dangerous one and should be dealt with with care, otherwise it may lead to nonsense like

$$ 1 = 1 ^ {1/2} = \ ( e ^ {2 \pi i } ) ^ {1/2} = \ e ^ {\pi i } = - 1. $$

By considering a single-valued branch of the logarithm (cf. Branch of an analytic function), or by considering the complete analytic function $ \mathop{\rm Ln} $ on its associated Riemann surface, an awkward notation and a lot of confusion may disappear. For fixed $ a \in \mathbf C \setminus 0 $, any value (i.e. determination) of $ \mathop{\rm Ln} a $ defines an exponential function:

$$ a ^ {z} = \ e ^ {z ( \textrm{ value } \textrm{ of } \mathop{\rm Ln} a) } . $$

References

[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
[a2] J.A. Dieudonné, "Foundations of modern analysis" , 1 , Acad. Press (1969) pp. 192 (Translated from French)
[a3] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
How to Cite This Entry:
Exponential function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exponential_function&oldid=46875
This article was adapted from an original article by Yu.V. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article