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Encyclopedia > Exponential function

The exponential function is one of the most important functions in mathematics. The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form ex, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point.
The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point.

As a function of the real variable x, the graph of y=ex is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Look up Slope in Wiktionary, the free dictionary. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...


Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form kax, where a, called the base, is any positive real number not equal to one. This article will focus initially on the exponential function with base e, Euler's number. Part of a scientific laboratory at the University of Cologne. ...


In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below. In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Contents

Properties

Most simply, exponential functions multiply at a constant rate. For example the population of a bacterial culture which doubles every 20 minutes can (approximatively, as this is not really a continuous problem) be expressed as an exponential, as can the value of a car which decreases by 10% per year.


Using the natural logarithm, one can define more general exponential functions. The function

,!, a^x=(e^{ln a})^x=e^{x ln a}

defined for all a > 0, and all real numbers x, is called the exponential function with base a. Note that this definition of , a^x rests on the previously established existence of the function , e^x , defined for all real numbers. (Here, we neither formally nor conceptually clarify whether such a function exists or what non-natural exponents are supposed to mean.)


Note that the equation above holds for a = e, since

,!, e^{x ln e}=e^{x cdot 1}=e^x.

Exponential functions "translate between addition and multiplication" as is expressed in the first three and the fifth of the following exponential laws:

,!, a^0 = 1
,!, a^1 = a
,!, a^{x + y} = a^x a^y
,!, a^{x y} = left( a^x right)^y
,!, {1 over a^x} = left({1 over a}right)^x = a^{-x}
,!, a^x b^x = (a b)^x

These are valid for all positive real numbers a and b and all real numbers x and y. Expressions involving fractions and roots can often be simplified using exponential notation: For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... In mathematics, an nth root of a number a is a number b, such that bn=a. ...

,{1 over a} = a^{-1}

and, for any a > 0, real number b, and integer n > 1:

,sqrt[n]{a^b} = left(sqrt[n]{a}right)^b = a^{b/n}.

Derivatives and differential equations

The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular, For a non-technical overview of the subject, see Calculus. ...

,{d over dx} e^x = e^x

That is, ex is its own derivative. Functions of the form ,Ke^x for constant K are the only functions with that property. (This follows from the Picard-Lindelöf theorem, with ,y(t) = e^t, y(0)=K and ,f(t,y(t)) = y(t).) Other ways of saying the same thing include: For a non-technical overview of the subject, see Calculus. ... In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ...

  • The slope of the graph at any point is the height of the function at that point.
  • The rate of increase of the function at x is equal to the value of the function at x.
  • The function solves the differential equation ,y'=y.
  • exp is a fixed point of derivative as a functional

In fact, many differential equations give rise to exponential functions, including the Schrödinger equation and the Laplace's equation as well as the equations for simple harmonic motion. A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ... In mathematics, the term functional is applied to certain functions. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ...


For exponential functions with other bases:

,{d over dx} a^x = (ln a) a^x

Thus any exponential function is a constant multiple of its own derivative. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...


If a variable's growth or decay rate is proportional to its size — as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay — then the variable can be written as a constant times an exponential function of time. In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio. ... Malthusian catastrophe, sometimes known as a Malthusian check, Malthusian crisis, Malthusian dilemma, Malthusian disaster, Malthusian trap, or Malthusian limit is a return to subsistence-level conditions as a result of agricultural (or, in later formulations, economic) production being eventually outstripped by growth in population[1]. Theories of Malthusian catastrophe are... It has been suggested that Interest expense be merged into this article or section. ... Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. ...


Furthermore for any differentiable function f(x), we find, by the chain rule: In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

,{d over dx} e^{f(x)} = f'(x)e^{f(x)}.

Formal definition

The exponential function (in blue), and the sum of the first n+1 terms of the power series on the left (in red).

The exponential function ex can be defined in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

e^x = sum_{n = 0}^{infty} {x^n over n!} = 1 + x + {x^2 over 2!} + {x^3 over 3!} + {x^4 over 4!} + cdots

or as the limit of a sequence: The limit of a sequence is one of the oldest concepts in mathematical analysis. ...

e^x = lim_{n to infty} left( 1 + {x over n} right)^n.

In these definitions, n! stands for the factorial of n, and x can be any real number, complex number, element of a Banach algebra (for example, a square matrix), or member of the field of p-adic numbers. For factorial rings in mathematics, see unique factorisation domain. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... For the square matrix section, see square matrix. ... The title given to this article is incorrect due to technical limitations. ...


For further explanation of these definitions and a proof of their equivalence, see the article Characterizations of the exponential function. In mathematics, the exponential function can be characterized in many ways. ...


Numerical value

To obtain the numerical value of the exponential function, the infinite series can be rewritten as :

,e^x = {1 over 0!} + x , left( {1 over 1!} + x , left( {1 over 2!} + x , left( {1 over 3!} + cdots fgright)right)right)
,= 1 + {x over 1} left(1 + {x over 2} left(1 + {x over 3} left(1 + cdots right)right)right)

This expression will converge quickly if we can ensure that x is less than one.


To ensure this, we can use the following identity.

,e^x, ,=e^{z+f},
,= e^z times left[{1 over 0!} + f , left( {1 over 1!} + f , left( {1 over 2!} + f , left( {1 over 3!} + cdots right)right)right)right]
  • Where ,z is the integer part of ,x
  • Where ,f is the fractional part of ,x
  • Hence, ,f is always less than 1 and ,f and ,z add up to ,x.

The value of the constant ez can be calculated beforehand by multiplying e with itself z times.


Computing exp(x) for real x

An even better algorithm can be found as follows.


First, notice that the answer y = ex is usually a floating point number represented by a mantissa m and an exponent n so y = m 2n for some integer n and suitably small m. Thus, we get: For the traditional use of the word mantissa in mathematics, see common logarithm. ...

,y = m,2^n = e^x.

Taking log on both sides of the last two gives us:

,ln(y) = ln(m) + nln(2) = x.

Thus, we get n as the result of dividing x by log(2) and finding the greatest integer that is not greater than this - that is, the floor function: The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ...

,n = leftlfloorfrac{x}{ln(2)}rightrfloor.

Having found n we can then find the fractional part u like this:

,u = x - nln(2).

The number u is small and in the range 0 ≤ u < ln(2) and so we can use the previously mentioned series to compute m:

,m = e^u = 1 + u(1 + u(frac{1}{2!} + u(frac{1}{3!} + u(....)))).

Having found m and n we can then produce y by simply combining those two into a floating point number:

,y = e^x = m,2^n.

Continued fractions for ex

Via Euler's identity:

,  e^x=1+x+frac{x^2}{2!}+cdots= 1+cfrac{x}{1-cfrac{x}{x+2-cfrac{2x}{x+3-cfrac{3x}{x+4-cfrac{4x}{x+5-cfrac{5x}{ddots}}}}}}

More advanced techniques are necessary to construct the following:

,  e^{2m/n}=1+cfrac{2m}{(n-m)+cfrac{m^2}{3n+cfrac{m^2}{5n+cfrac{m^2}{7n+cfrac{m^2}{9n+cfrac{m^2}{ddots}}}}}},

Setting m = x and n = 2 yields

,  e^x=1+cfrac{2x}{(2-x)+cfrac{x^2}{6+cfrac{x^2}{10+cfrac{x^2}{14+cfrac{x^2}{18+cfrac{x^2}{ddots}}}}}},

Computation of ,a^n for natural number (positive integer) n

There is a fast way to compute ,a^n when n is a positive integer. It makes use of the fact that testing that such a number is odd is very easy on a computer and dividing by 2 is also fast by simply shifting all the bits to the right.


step 1, initialize some variables
y := 1, k := n, f := a


step 2, test k
if k is 0, go to step 7


step 3, (k is not 0 here, test if k is even)
if k is even go to step 5


step 4, (k is odd here, multiply in)
, y := y * f


step 5, (divide k by 2 / ignore remainder, divide by shift, also square f)
k := k shift right by 1 f := f * f


step 6, (loop)
go back to step 2


step 7, (done, y is result = an)
return y


In C you can write the algorithm like this: C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ...

  double power(double a, unsigned int n) { double y = 1; double f = a; unsigned int k = n; while (k != 0) { if ((k & 1) != 0) y *= f; k >>= 1; f *= f; } return y; }  

While a naive multiplication of a^100 would require 100 iterations of a loop multiplying a, this loop iterates only 7 times (The number 100 is written using 7 bits).


This algorithm can easily be extended for signed integers by doing the following steps before and after:


step 1. if k is negative, negate the value so we get a positive k. n still remembers the original value.


step 2. Perform the above computation for ,y = a^{|k|}


step 3. If n is negative, invert the result so y := 1/y. y is now the result of ,a^n for an integer n.


On the complex plane

Exponential function on the complex plane. The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. The periodic horizontal bands indicate that the exponential function is periodic in the imaginary part of its argument.
Exponential function on the complex plane. The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. The periodic horizontal bands indicate that the exponential function is periodic in the imaginary part of its argument.

When considered as a function defined on the complex plane, the exponential function retains the important properties Image File history File links Metadata Size of this preview: 600 × 600 pixelsFull resolution (651 × 651 pixel, file size: 128 KB, MIME type: image/jpeg) This is the color function used in the picture above File historyClick on a date/time to view the file as it appeared at that... Image File history File links Metadata Size of this preview: 600 × 600 pixelsFull resolution (651 × 651 pixel, file size: 128 KB, MIME type: image/jpeg) This is the color function used in the picture above File historyClick on a date/time to view the file as it appeared at that... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

,!, e^{z + w} = e^z e^w
,!, e^0 = 1
,!, e^z ne 0
,!, {d over dz} e^z = e^z

for all z and w.


It is a holomorphic function which is periodic with imaginary period ,2 pi i and can be written as Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

,!, e^{a + bi} = e^a (cos b + i sin b)

where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another. In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ &#8722; × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...


See also Euler's formula. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...


Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation: This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...

,!, z^w = e^{w ln z}

for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.


The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases might be noted: when the original line is parallel to the real axis, the resulting sprial never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius. “Line” redirects here. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

Image File history File links Size of this preview: 762 × 600 pixel Image in higher resolution (1250 × 984 pixel, file size: 584 KB, MIME type: image/png) Graphs of z = Re(e^(x+iy)), z = Im(e^(x+iy)) and z = |e^(x+iy)|, selfmade with MuPad. ...

Computation of exp(z) for a complex z

This is fairly straightforward given the formula

,e^{x + yi} = e^xe^{yi} = e^x(cos(y) + i sin(y)) = e^xcos(y) + ie^xsin(y).

Note that the argument y to the trigonometric functions is real.


Computation of ,a^b where both a and b are complex

This is also straightforward given the formulae:


if a = x + yi and b = u + vi we can first convert a to polar co-ordinates by finding a ,theta and an ,r such that:

,re^{{theta}i} = rcostheta + i rsintheta = a = x + yi

or

, x = rcostheta and ,y = rsintheta.

Thus, ,x^2 + y^2 = r^2 or ,r = sqrt{x^2 + y^2} and ,tantheta = frac{y}{x} or ,theta = operatorname{atan2}(y, x).


Now, we have that:

,a = re^{{theta}i} = e^{log(r) + {theta}i}

so:

,a^b = (e^{log(r) + {theta}i})^{u + vi} = e^{(log(r) + {theta}i)(u + vi)}

The exponent is thus a simple multiplication of two complex values yielding a complex result which can then be brought back to regular cartesian format by the formula:

,e^{p + qi} = e^p(cos(q) + isin(q)) = e^pcos(q) + ie^psin(q)

where p is the real part of the multiplication:

,p = ulog(r) - vtheta

and q is the imaginary part of the multiplication:

,q = vlog(r) + utheta.

Note that all of ,x, y, u, v, r, ,theta, ,p and ,q are all real values in these computations. The result of ,a^b is thus ,p + qi.


Also note that since we compute and use ,log(r) rather than r itself you don't have to compute the square root. Instead simply compute ,log(r) = frac12log(x^2 + y^2). Watch out for potential overflow though and possibly scale down the x and y prior to computing ,x^2 + y^2 by a suitable power of 2 if ,x and ,y are so large that you would overflow. If you instead run the risk of underflow, scale up by a suitable power of 2 prior to computing the sum of the squares. In either case you then get the scaled version of ,x - we can call it ,x' and the scaled version of ,y - call it ,y' and so you get:

,x = x'2^s and ,y = y'2^s

where ,2^s is the scaling factor.


Then you get ,log(r) = frac12(log(x'^2 + y'^2) + s) where ,x' and ,y' are scaled so that the sum of the squares will not overflow or underflow. If ,x is very large while ,y is very small so that you cannot find such a scaling factor you will overflow anyway and so the sum is essentially equal to ,x^2 since y is ignored and thus you get ,r = |x| in this case and ,log(r) = log(|x|). The same happens in the case when ,x is very small and ,y is very large. If both are very large or both are very small you can find a scaling factor as mentioned earlier.


Note that this function is, in general, multivalued for complex arguments. This is because rotation of a single point through any angle plus 360 degrees, or radians, is the same as rotation through the angle itself. So θ above is not unique: θk = θ + 2πk for any integer k would do as well. The convention though is that when ab is taken as a single value it must be that for k = 0, ie. we use the smallest possible (in magnitude) value of theta, which has a magnitude of, at most, π. This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...


Matrices and Banach algebras

The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential). In this case we have In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ...

, e^{x + y} = e^x e^y mbox{ if } xy = yx
, e^0 = 1
, e^x is invertible with inverse , e^{-x}
the derivative of , e^x at the point , x is that linear map which sends , u to , ue^x.

In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument: In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

, f(t) = e^{t A}

where A is a fixed element of the algebra and t is any real number. This function has the important properties

, f(s + t) = f(s) f(t)
, f(0) = 1
, f'(t) = A f(t)

On Lie algebras

The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M (n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ...


Double exponential function

The term double exponential function can have two meanings: A double exponential function is a constant raised to the power of an exponential function. ...

  • a function with two exponential terms, with different exponents
  • a function ,f(x) = a^{a^x}; this grows even faster than an exponential function; for example, if a = 10: f(−1) = 1.26, f(0) = 10, f(1) = 1010, f(2) = 10100 = googol, ..., f(100) = googolplex.

Factorials grow faster than exponential functions, but slower than double-exponential functions. Fermat numbers, generated by ,F(m) = 2^{2^m} + 1 and double Mersenne numbers generated by ,MM(p) = 2^{(2^p-1)}-1 are examples of double exponential functions. Not to be confused with Google, the Internet company, and Nikolai Gogol, the author. ... This article is about a number. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, a double Mersenne number is a Mersenne number of the form where n is a positive integer. ...


Similar properties of e and the function ez

The function ez is not in C(z) (ie. not the quotient of two polynomials with complex coefficients).


For n distinct complex numbers {a1,...an}, {e^{a_1 z},... e^{a_n z}} is linearly independent over C(z).


The function ez is transcendental over C(z).


See also

Mathematics Portal

Image File history File links Portal. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, the exponential function can be characterized in many ways. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... The following is a list of integrals (antiderivative functions) of exponential functions. ... This is a list of exponential topics, by Wikipedia page. ...

External links


  Results from FactBites:
 
Exponential function - Wikipedia, the free encyclopedia (1112 words)
The exponential function is one of the most important functions in mathematics.
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
Encyclopedia4U - Exponential function - Encyclopedia Article (732 words)
3 Exponential function for matrices and Banach algebras
This formula connects the exponential function with the trigonometric functions, and this is the reason that extending the natural logarithm to complex arguments yields a multi-valued function ln(z).
It is easy to see, that the exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the centre at 0, noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.
  More results at FactBites »

 
 

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