86-423: Exp or EXP may stand for: Exponential function , in mathematics Expiry date of organic compounds like food or medicines Experience points , in role-playing games EXPTIME , a complexity class in computing Ford EXP , a car manufactured in the 1980s Exp (band) , an Italian group in the 1990s "EXP" (song) , a song by The Jimi Hendrix Experience from
172-443: A e x ln ( b ) ln ( b ) = a b x ln ( b ) . {\displaystyle {\frac {d}{dx}}a\,b^{x}={\frac {d}{dx}}a\,e^{x\ln(b)}=a\,e^{x\ln(b)}\ln(b)=a\,b^{x}\ln(b).} Let a > 0 {\displaystyle a>0} be a positive coefficient. For b > 1 {\displaystyle b>1} ,
258-1667: A e k x {\displaystyle f(x)=ae^{kx}} for some constant a . The constant k is called the decay constant , disintegration constant , rate constant , or transformation constant . Furthermore, for any differentiable function f , we find, by the chain rule : d d x e f ( x ) = f ′ ( x ) e f ( x ) . {\displaystyle {\frac {d}{dx}}e^{f(x)}=f'(x)\,e^{f(x)}.} A continued fraction for e can be obtained via an identity of Euler : e x = 1 + x 1 − x x + 2 − 2 x x + 3 − 3 x x + 4 − ⋱ {\displaystyle e^{x}=1+{\cfrac {x}{1-{\cfrac {x}{x+2-{\cfrac {2x}{x+3-{\cfrac {3x}{x+4-\ddots }}}}}}}}} The following generalized continued fraction for e converges more quickly: e z = 1 + 2 z 2 − z + z 2 6 + z 2 10 + z 2 14 + ⋱ {\displaystyle e^{z}=1+{\cfrac {2z}{2-z+{\cfrac {z^{2}}{6+{\cfrac {z^{2}}{10+{\cfrac {z^{2}}{14+\ddots }}}}}}}}} or, by applying
344-404: A cubic function with a local maximum and a local minimum has three branches (see the adjacent picture). These considerations are particularly important for defining the inverses of trigonometric functions . For example, the sine function is not one-to-one, since for every real x (and more generally sin( x + 2 π n ) = sin( x ) for every integer n ). However, the sine is one-to-one on
430-607: A function f : X → Y {\displaystyle f\colon X\to Y} , its inverse f − 1 : Y → X {\displaystyle f^{-1}\colon Y\to X} admits an explicit description: it sends each element y ∈ Y {\displaystyle y\in Y} to the unique element x ∈ X {\displaystyle x\in X} such that f ( x ) = y . As an example, consider
516-406: A function f : X → X with itself is called iteration . If f is applied n times, starting with the value x , then this is written as f ( x ) ; so f ( x ) = f ( f ( x )) , etc. Since f ( f ( x )) = x , composing f and f yields f , "undoing" the effect of one application of f . While the notation f ( x ) might be misunderstood, ( f ( x )) certainly denotes
602-565: A function of the form f ( x ) = a b c x + d {\displaystyle f(x)=ab^{cx+d}} is also an exponential function: a b c x + d = ( a b d ) ( b c ) x = ( a b d ) e x c ln b . {\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}=\left(ab^{d}\right)e^{xc\ln b}.} The exponential function arises whenever
688-518: A function whose domain is the set X , and whose codomain is the set Y . Then f is invertible if there exists a function g from Y to X such that g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for all x ∈ X {\displaystyle x\in X} and f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for all y ∈ Y {\displaystyle y\in Y} . If f
774-519: A neighborhood of a point p as long as the Jacobian matrix of f at p is invertible . In this case, the Jacobian of f at f ( p ) is the matrix inverse of the Jacobian of f at p . Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function is not one-to-one, since x = (− x ) . However,
860-483: A quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest , and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ ( 1 + 1 n ) n {\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}} now known as e . Later, in 1697, Johann Bernoulli studied
946-429: A real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b : d d x a b x = d d x a e x ln ( b ) =
SECTION 10
#17327729122961032-1747: A special case for z = 2 : e 2 = 1 + 4 0 + 2 2 6 + 2 2 10 + 2 2 14 + ⋱ = 7 + 2 5 + 1 7 + 1 9 + 1 11 + ⋱ {\displaystyle e^{2}=1+{\cfrac {4}{0+{\cfrac {2^{2}}{6+{\cfrac {2^{2}}{10+{\cfrac {2^{2}}{14+\ddots \,}}}}}}}}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots \,}}}}}}}}} This formula also converges, though more slowly, for z > 2 . For example: e 3 = 1 + 6 − 1 + 3 2 6 + 3 2 10 + 3 2 14 + ⋱ = 13 + 54 7 + 9 14 + 9 18 + 9 22 + ⋱ {\displaystyle e^{3}=1+{\cfrac {6}{-1+{\cfrac {3^{2}}{6+{\cfrac {3^{2}}{10+{\cfrac {3^{2}}{14+\ddots \,}}}}}}}}=13+{\cfrac {54}{7+{\cfrac {9}{14+{\cfrac {9}{18+{\cfrac {9}{22+\ddots \,}}}}}}}}} As in
1118-413: Is injective , and the condition f ( g ( y ) ) = y {\displaystyle f(g(y))=y} for all y ∈ Y {\displaystyle y\in Y} implies that f is surjective . The inverse function f to f can be explicitly described as the function Recall that if f is an invertible function with domain X and codomain Y , then Using
1204-571: Is a conformal map from an infinite strip of the complex plane − π < Im z ≤ π {\displaystyle -\pi <\operatorname {Im} z\leq \pi } (which periodically repeats in the imaginary direction) onto the whole complex plane except for 0 {\displaystyle 0} . The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras . Some old texts refer to
1290-419: Is a consequence of the implication that for f to be invertible it must be bijective. The involutory nature of the inverse can be concisely expressed by The inverse of a composition of functions is given by Notice that the order of g and f have been reversed; to undo f followed by g , we must first undo g , and then undo f . For example, let f ( x ) = 3 x and let g ( x ) = x + 5 . Then
1376-401: Is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718 , the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature. The exponential function converts sums to products: it maps the additive identity 0 to the multiplicative identity 1 , and
1462-723: Is called the (positive) square root function and is denoted by x ↦ x {\displaystyle x\mapsto {\sqrt {x}}} . The following table shows several standard functions and their inverses: Many functions given by algebraic formulas possess a formula for their inverse. This is because the inverse f − 1 {\displaystyle f^{-1}} of an invertible function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } has an explicit description as This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, if f
1548-404: Is called the "natural exponential function", or simply "the exponential function", denoted as x ↦ e x or x ↦ exp ( x ) . {\displaystyle x\mapsto e^{x}\quad {\text{or}}\quad x\mapsto \exp(x).} The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent
1634-476: Is compounded daily, this becomes (1 + x / 365 ) . Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp x = lim n → ∞ ( 1 + x n ) n {\displaystyle \exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}} first given by Leonhard Euler . This
1720-716: Is decreasing (as depicted for b = 1 / 2 ), and describes exponential decay. For b = 1 , the function is constant. Euler's number e = 2.71828... is the unique base for which the constant of proportionality is 1, since ln ( e ) = 1 {\displaystyle \ln(e)=1} , so that the function is its own derivative: d d x e x = e x ln ( e ) = e x . {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\ln(e)=e^{x}.} This function, also denoted as exp ( x ) {\displaystyle \exp(x)} ,
1806-480: Is either continuous or monotonic Positiveness: For every x {\displaystyle x} , one has exp ( x ) ≠ 0 {\displaystyle \exp(x)\neq 0} , since the functional equation implies exp ( x ) exp ( − x ) = 1 {\displaystyle \exp(x)\exp(-x)=1} . It results that
SECTION 20
#17327729122961892-403: Is either strictly increasing or decreasing (with no local maxima or minima ). For example, the function is invertible, since the derivative f′ ( x ) = 3 x + 1 is always positive. If the function f is differentiable on an interval I and f′ ( x ) ≠ 0 for each x ∈ I , then the inverse f is differentiable on f ( I ) . If y = f ( x ) , the derivative of
1978-400: Is equal to id X . Such a function is called an involution . If f is invertible, then the graph of the function is the same as the graph of the equation This is identical to the equation y = f ( x ) that defines the graph of f , except that the roles of x and y have been reversed. Thus the graph of f can be obtained from the graph of f by switching the positions of
2064-470: Is equal to 1 when x = 0 . That is, d d x e x = e x and e 0 = 1. {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\quad {\text{and}}\quad e^{0}=1.} Functions of the form ae for constant a are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem ). Other ways of saying
2150-516: Is invertible, then there is exactly one function g satisfying this property. The function g is called the inverse of f , and is usually denoted as f , a notation introduced by John Frederick William Herschel in 1813. The function f is invertible if and only if it is bijective. This is because the condition g ( f ( x ) ) = x {\displaystyle g(f(x))=x} for all x ∈ X {\displaystyle x\in X} implies that f
2236-439: Is more complicated and harder to read in a small font. Since any exponential function f ( x ) = a b x {\displaystyle f(x)=a\,b^{x}} can be written in terms of the natural exponential, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. For real numbers c , d {\displaystyle c,d} ,
2322-473: Is one of a number of characterizations of the exponential function ; others involve series or differential equations . From any of these definitions it can be shown that e is the reciprocal of e . For example, from the differential equation definition, e e = 1 when x = 0 and its derivative using the product rule is e e − e e = 0 for all x , so e e = 1 for all x . From any of these definitions it can be shown that
2408-603: Is positive or exponential decay if c {\displaystyle c} is negative. The exponential function can be generalized to accept a complex number as its argument. This reveals a relation between the multiplication of complex numbers and rotation in the Euclidean plane , Euler's formula exp i θ = cos θ + i sin θ {\displaystyle \exp i\theta =\cos \theta +i\sin \theta } :
2494-473: Is sometimes called the natural exponential function to distinguish it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive b , a b x = def a e x ln b {\displaystyle a\,b^{x}\mathrel {\stackrel {\text{def}}{=}} a\,e^{x\ln b}} As functions of
2580-406: Is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every x {\displaystyle x} per the ratio test . So, the derivative of the sum can be computed by term-by-term derivation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that
2666-421: Is the inverse function of the natural logarithm . The inverse function theorem implies that the natural logarithm has an inverse function, that satisfies the above definition. This is a first proof of existence. Therefore, one has for every real number x {\displaystyle x} and evey positive real number y . {\displaystyle y.} The exponential function
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2752-415: Is the function then f is a bijection, and therefore possesses an inverse function f . The formula for this inverse has an expression as an infinite sum: Since a function is a special type of binary relation , many of the properties of an inverse function correspond to properties of converse relations . If an inverse function exists for a given function f , then it is unique. This follows since
2838-419: Is the function then to determine f − 1 ( y ) {\displaystyle f^{-1}(y)} for a real number y , one must find the unique real number x such that (2 x + 8) = y . This equation can be solved: Thus the inverse function f is given by the formula Sometimes, the inverse of a function cannot be expressed by a closed-form formula . For example, if f
2924-571: Is the sum of a power series : exp ( x ) = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ = ∑ n = 0 ∞ x n n ! , {\displaystyle {\begin{aligned}\exp(x)&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}},\end{aligned}}} where n ! {\displaystyle n!}
3010-420: Is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable x {\displaystyle x} is denoted exp x {\displaystyle \exp x} or e x {\displaystyle e^{x}} , with the two notations used interchangeably. It
3096-504: The Cauchy sense, permitted by Mertens' theorem , shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments: exp ( w + z ) = exp w exp z for all w , z ∈ C {\displaystyle \exp(w+z)=\exp w\exp z{\text{ for all }}w,z\in \mathbb {C} } The definition of
3182-429: The complex plane . Euler's formula relates its values at purely imaginary arguments to trigonometric functions . The exponential function also has analogues for which the argument is a matrix , or even an element of a Banach algebra or a Lie algebra . The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and
3268-416: The composition of functions , this statement can be rewritten to the following equations between functions: where id X is the identity function on the set X ; that is, the function that leaves its argument unchanged. In category theory , this statement is used as the definition of an inverse morphism . Considering function composition helps to understand the notation f . Repeatedly composing
3354-429: The multiplicative inverse of f ( x ) and has nothing to do with the inverse function of f . The notation f ⟨ − 1 ⟩ {\displaystyle f^{\langle -1\rangle }} might be used for the inverse function to avoid ambiguity with the multiplicative inverse . In keeping with the general notation, some English authors use expressions like sin ( x ) to denote
3440-489: The ratio test and are therefore entire functions (that is, holomorphic on C {\displaystyle \mathbb {C} } ). The range of the exponential function is C ∖ { 0 } {\displaystyle \mathbb {C} \setminus \{0\}} , while the ranges of the complex sine and cosine functions are both C {\displaystyle \mathbb {C} } in its entirety, in accord with Picard's theorem , which asserts that
3526-513: The real case, the exponential function can be defined on the complex plane in several equivalent forms. The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: exp z := ∑ k = 0 ∞ z k k ! {\displaystyle \exp z:=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}} Alternatively,
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3612-630: The real-valued function of a real variable given by f ( x ) = 5 x − 7 . One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of f is the function f − 1 : R → R {\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} } defined by f − 1 ( y ) = y + 7 5 . {\displaystyle f^{-1}(y)={\frac {y+7}{5}}.} Let f be
3698-411: The slope of the tangent to the graph at each point is equal to its height (its y -coordinate) at that point. There are several different definitions of the exponential function, which are all equivalent, although of very dirrerent nature. One of the simplest definitions is: The exponential function is the unique differentiable function that equals its derivative , and takes the value 1 for
3784-439: The x and y axes. This is equivalent to reflecting the graph across the line y = x . By the inverse function theorem , a continuous function of a single variable f : A → R {\displaystyle f\colon A\to \mathbb {R} } (where A ⊆ R {\displaystyle A\subseteq \mathbb {R} } ) is invertible on its range (image) if and only if it
3870-456: The above definition, then the derivative of f / g {\displaystyle f/g} is zero everywhere by the quotient rule . It follows that g ( x ) {\displaystyle g(x)} is constant, and this constant is 1 since f ( 0 ) = g ( 0 ) = 1 {\displaystyle f(0)=g(0)=1} . The exponential function
3956-414: The album Axis: Bold as Love EXP (calculator key) , to enter numbers in scientific or engineering notation EXP , at Northeastern University See also [ edit ] Exponential map (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Exp . If an internal link led you here, you may wish to change
4042-407: The ambiguity of the f notation should be avoided. The function f : R → [0,∞) given by f ( x ) = x is not injective because ( − x ) 2 = x 2 {\displaystyle (-x)^{2}=x^{2}} for all x ∈ R {\displaystyle x\in \mathbb {R} } . Therefore, f is not invertible. If the domain of
4128-518: The balance of a bank account bearing compound interest , the size of a bacterial population, the temperature of an object relative to its environment, or the amount of a radioactive substance, can be modeled using functions of the form y ( t ) = a e c t {\displaystyle y(t)=ae^{ct}} , also sometimes called exponential functions ; these quantities undergo exponential growth if c {\displaystyle c}
4214-421: The calculus of the exponential function. If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x / 12 times the current value, so each month the total value is multiplied by (1 + x / 12 ) , and the value at the end of the year is (1 + x / 12 ) . If instead interest
4300-463: The complex exponential function based on this relationship. If z = x + iy , where x and y are both real, then we could define its exponential as exp z = exp ( x + i y ) := ( exp x ) ( cos y + i sin y ) {\displaystyle \exp z=\exp(x+iy):=(\exp x)(\cos y+i\sin y)} where exp , cos , and sin on
4386-909: The complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments. In particular, when z = it ( t real), the series definition yields the expansion exp ( i t ) = ( 1 − t 2 2 ! + t 4 4 ! − t 6 6 ! + ⋯ ) + i ( t − t 3 3 ! + t 5 5 ! − t 7 7 ! + ⋯ ) . {\displaystyle \exp(it)=\left(1-{\frac {t^{2}}{2!}}+{\frac {t^{4}}{4!}}-{\frac {t^{6}}{6!}}+\cdots \right)+i\left(t-{\frac {t^{3}}{3!}}+{\frac {t^{5}}{5!}}-{\frac {t^{7}}{7!}}+\cdots \right).} In this expansion,
SECTION 50
#17327729122964472-489: The complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one: exp z := lim n → ∞ ( 1 + z n ) n {\displaystyle \exp z:=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}} For the power series definition, term-wise multiplication of two copies of this power series in
4558-420: The composition g ∘ f is the function that first multiplies by three and then adds five, To reverse this process, we must first subtract five, and then divide by three, This is the composition ( f ∘ g )( x ) . If X is a set, then the identity function on X is its own inverse: More generally, a function f : X → X is equal to its own inverse, if and only if the composition f ∘ f
4644-453: The equivalence of the two notations for the exponential function. The exponential function is the limit exp ( x ) = lim n → ∞ ( 1 + x n ) n , {\displaystyle \exp(x)=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n},} where n {\displaystyle n} takes only integer values (otherwise,
4730-1213: The equivalent power series: cos z := exp ( i z ) + exp ( − i z ) 2 = ∑ k = 0 ∞ ( − 1 ) k z 2 k ( 2 k ) ! , and sin z := exp ( i z ) − exp ( − i z ) 2 i = ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! {\displaystyle {\begin{aligned}&\cos z:={\frac {\exp(iz)+\exp(-iz)}{2}}=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k}}{(2k)!}},\\[5pt]{\text{and }}\quad &\sin z:={\frac {\exp(iz)-\exp(-iz)}{2i}}=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k+1}}{(2k+1)!}}\end{aligned}}} for all z ∈ C . {\textstyle z\in \mathbb {C} .} The functions exp , cos , and sin so defined have infinite radii of convergence by
4816-405: The existence of some y {\displaystyle y} such that exp ( y ) = 0 {\displaystyle \exp(y)=0} . It results also that the exponential function is monotonically increasing . Extension of exponentiation to positive real bases: Let b be a positive real number. The exponential function and
4902-798: The exponential and natural logarithm functions, as b x = exp ( x ⋅ ln b ) {\displaystyle b^{x}=\exp(x\cdot \ln b)} . The "natural" base e = exp 1 {\displaystyle e=\exp 1} is the unique base satisfying the criterion that the exponential function's derivative equals its value, d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} , which simplifies definitions and eliminates extraneous constants when using exponential functions in calculus . Quantities which change over time in proportion to their value, for example
4988-513: The exponential function as the antilogarithm . The graph of y = e x {\displaystyle y=e^{x}} is upward-sloping, and increases faster as x increases. The graph always lies above the x -axis, but becomes arbitrarily close to it for large negative x ; thus, the x -axis is a horizontal asymptote . The equation d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that
5074-442: The exponential function is defined for every x {\displaystyle x} , and is everywhere the sum of its Maclaurin series . The exponential satisfies the functional equation: exp ( x + y ) = exp ( x ) ⋅ exp ( y ) . {\displaystyle \exp(x+y)=\exp(x)\cdot \exp(y).} This results from
5160-411: The exponential function is positive (since exp ( 0 ) > 0 {\displaystyle \exp(0)>0} , if one would have exp ( x ) < 0 {\displaystyle \exp(x)<0} for some x {\displaystyle x} , the intermediate value theorem would imply
5246-1208: The exponential function obeys the basic exponentiation identity. For example, from the power series definition, expanded by the Binomial theorem , exp ( x + y ) = ∑ n = 0 ∞ ( x + y ) n n ! = ∑ n = 0 ∞ ∑ k = 0 n n ! k ! ( n − k ) ! x k y n − k n ! = ∑ k = 0 ∞ ∑ ℓ = 0 ∞ x k y ℓ k ! ℓ ! = exp x ⋅ exp y . {\displaystyle \exp(x+y)=\sum _{n=0}^{\infty }{\frac {(x+y)^{n}}{n!}}=\sum _{n=0}^{\infty }\sum _{k=0}^{n}{\frac {n!}{k!(n-k)!}}{\frac {x^{k}y^{n-k}}{n!}}=\sum _{k=0}^{\infty }\sum _{\ell =0}^{\infty }{\frac {x^{k}y^{\ell }}{k!\ell !}}=\exp x\cdot \exp y\,.} This justifies
SECTION 60
#17327729122965332-404: The exponential function of base e {\displaystyle e} is occasionally called the natural exponential function , matching the name natural logarithm . The generalization of the standard exponent notation b x {\displaystyle b^{x}} to arbitrary real numbers as exponents, is usually formally defined in terms of
5418-423: The exponential notation e for exp x . The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function. This derivative property leads to exponential growth or exponential decay. The exponential function extends to an entire function on
5504-699: The exponential of a sum is equal to the product of separate exponentials, exp ( x + y ) = exp x ⋅ exp y {\displaystyle \exp(x+y)=\exp x\cdot \exp y} . Its inverse function , the natural logarithm , ln {\displaystyle \ln } or log {\displaystyle \log } , converts products to sums: ln ( x ⋅ y ) = ln x + ln y {\displaystyle \ln(x\cdot y)=\ln x+\ln y} . Other functions of
5590-521: The exponential of an imaginary number i θ {\displaystyle i\theta } is a point on the complex unit circle at angle θ {\displaystyle \theta } from the real axis. The identities of trigonometry can thus be translated into identities involving exponentials of imaginary quantities. The complex function z ↦ exp z {\displaystyle z\mapsto \exp z}
5676-720: The exponentiation would require the xponential function to be defined). By continuity of the logarithm, this can be proved by taking logarithms and proving x = lim n → ∞ ln ( 1 + x n ) n = lim n → ∞ n ln ( 1 + x n ) , {\displaystyle x=\lim _{n\to \infty }\ln \left(1+{\frac {x}{n}}\right)^{n}=\lim _{n\to \infty }n\ln \left(1+{\frac {x}{n}}\right),} for example with Taylor's theorem . The exponential function f ( x ) = e x {\displaystyle f(x)=e^{x}}
5762-411: The function a b x {\displaystyle a\,b^{x}} is increasing (as depicted for b = e and b = 2 ), because ln b > 0 {\displaystyle \ln b>0} makes the derivative always positive, and describes exponential growth. For 0 < b < 1 {\displaystyle 0<b<1} , the function
5848-399: The function becomes one-to-one if we restrict to the domain x ≥ 0 , in which case (If we instead restrict to the domain x ≤ 0 , then the inverse is the negative of the square root of y .) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function : Sometimes, this multivalued inverse is called the full inverse of f , and
5934-431: The function is restricted to the nonnegative reals, that is, we take the function f : [ 0 , ∞ ) → [ 0 , ∞ ) ; x ↦ x 2 {\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}} with the same rule as before, then the function is bijective and so, invertible. The inverse function here
6020-521: The general form f ( x ) = b x {\displaystyle f(x)=b^{x}} , with base b {\displaystyle b} , are also commonly called exponential functions , and share the property of converting addition to multiplication, b x + y = b x ⋅ b y {\displaystyle b^{x+y}=b^{x}\cdot b^{y}} . Where these two meanings might be confused,
6106-422: The inverse function must be the converse relation, which is completely determined by f . There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X and codomain Y , then its inverse f has domain Y and image X , and the inverse of f is the original function f . In symbols, for functions f : X → Y and f : Y → X , This statement
6192-415: The inverse is given by the inverse function theorem, Using Leibniz's notation the formula above can be written as This result follows from the chain rule (see the article on inverse functions and differentiation ). The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable multivariable function f : R → R is invertible in
6278-399: The inverse of the sine function applied to x (actually a partial inverse ; see below). Other authors feel that this may be confused with the notation for the multiplicative inverse of sin ( x ) , which can be denoted as (sin ( x )) . To avoid any confusion, an inverse trigonometric function is often indicated by the prefix " arc " (for Latin arcus ). For instance, the inverse of
6364-409: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Exp&oldid=1244611742 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Exponential function In mathematics , the exponential function
6450-500: The naturel logarithm being the inverse each of the other, one has b = exp ( ln b ) . {\displaystyle b=\exp(\ln b).} If n is an integer, the functional equation of the logarithm implies b n = exp ( ln b n ) = exp ( n ln b ) . {\displaystyle b^{n}=\exp(\ln b^{n})=\exp(n\ln b).} Since
6536-412: The partial inverse: sin − 1 ( x ) = { ( − 1 ) n arcsin ( x ) + π n : n ∈ Z } {\displaystyle \sin ^{-1}(x)=\{(-1)^{n}\arcsin(x)+\pi n:n\in \mathbb {Z} \}} . Other inverse special functions are sometimes prefixed with the prefix "inv", if
6622-402: The portions (such as √ x and − √ x ) are called branches . The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch , and its value at y is called the principal value of f ( y ) . For a continuous function on the real line, one branch is required between each pair of local extrema . For example, the inverse of
6708-616: The range of a nonconstant entire function is either all of C {\displaystyle \mathbb {C} } , or C {\displaystyle \mathbb {C} } excluding one lacunary value . These definitions for the exponential and trigonometric functions lead trivially to Euler's formula : exp ( i z ) = cos z + i sin z for all z ∈ C . {\displaystyle \exp(iz)=\cos z+i\sin z{\text{ for all }}z\in \mathbb {C} .} We could alternatively define
6794-478: The rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of cos t and sin t , respectively. This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of exp ( ± i z ) {\displaystyle \exp(\pm iz)} and
6880-491: The right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means. Inverse function In mathematics , the inverse function of a function f (also called the inverse of f ) is a function that undoes the operation of f . The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.} For
6966-812: The right-most expression is defined if n is any real number, this allows defining b x {\displaystyle b^{x}} for every positive real number b and every real number x : b x = exp ( x ln b ) . {\displaystyle b^{x}=\exp(x\ln b).} In particular, if b is the Euler's number e = exp ( 1 ) , {\displaystyle e=\exp(1),} one has ln e = 1 {\displaystyle \ln e=1} (inverse function) and thus e x = exp ( x ) . {\displaystyle e^{x}=\exp(x).} This shows
7052-530: The same thing include: 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. More generally, for any real constant k , a function f : R → R satisfies f ′ = k f {\displaystyle f'=kf} if and only if f ( x ) =
7138-400: The sine function is typically called the arcsine function, written as arcsin ( x ) . Similarly, the inverse of a hyperbolic function is indicated by the prefix " ar " (for Latin ārea ). For instance, the inverse of the hyperbolic sine function is typically written as arsinh ( x ) . The expressions like sin ( x ) can still be useful to distinguish the multivalued inverse from
7224-588: The substitution z = x / y : e x y = 1 + 2 x 2 y − x + x 2 6 y + x 2 10 y + x 2 14 y + ⋱ {\displaystyle e^{\frac {x}{y}}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+\ddots }}}}}}}}} with
7310-402: The uniqueness and the fact that the function f ( x ) = exp ( x + y ) / exp ( y ) {\displaystyle f(x)=\exp(x+y)/\exp(y)} satisfies the above definition. It can be proved that a function that satisfies this functional equation is the exponential function if its derivative at 0 is 1 and the function
7396-410: The value 0 of its variable. This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function. Uniqueness: If f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are two functions satisfying
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