In the mathematical field of complex analysis , contour integration is a method of evaluating certain integrals along paths in the complex plane .
115-624: In mathematics, the error function (also called the Gauss error function ), often denoted by erf , is a function e r f : C → C {\displaystyle \mathrm {erf} :\mathbb {C} \to \mathbb {C} } defined as: erf z = 2 π ∫ 0 z e − t 2 d t . {\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.} The integral here
230-748: A + ln ( 1 − x 2 ) 2 ) 2 − ln ( 1 − x 2 ) a − ( 2 π a + ln ( 1 − x 2 ) 2 ) . {\displaystyle \operatorname {erf} ^{-1}x\approx \operatorname {sgn} x\cdot {\sqrt {{\sqrt {\left({\frac {2}{\pi a}}+{\frac {\ln \left(1-x^{2}\right)}{2}}\right)^{2}-{\frac {\ln \left(1-x^{2}\right)}{a}}}}-\left({\frac {2}{\pi a}}+{\frac {\ln \left(1-x^{2}\right)}{2}}\right)}}.} The complementary error function , denoted erfc ,
345-516: A − μ 2 σ ) . {\displaystyle {\begin{aligned}\Pr[L_{a}\leq X\leq L_{b}]&=\int _{L_{a}}^{L_{b}}{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,\mathrm {d} x\\&={\frac {1}{2}}\left(\operatorname {erf} {\frac {L_{b}-\mu }{{\sqrt {2}}\sigma }}-\operatorname {erf} {\frac {L_{a}-\mu }{{\sqrt {2}}\sigma }}\right).\end{aligned}}} The property erf (− z ) = −erf z means that
460-640: A a f ( z ) d z = ∮ C f ( z ) d z − ∫ Arc f ( z ) d z {\displaystyle \int _{-a}^{a}f(z)\,dz=\oint _{C}f(z)\,dz-\int _{\text{Arc}}f(z)\,dz} Furthermore, observe that f ( z ) = 1 ( z 2 + 1 ) 2 = 1 ( z + i ) 2 ( z − i ) 2 . {\displaystyle f(z)={\frac {1}{\left(z^{2}+1\right)^{2}}}={\frac {1}{(z+i)^{2}(z-i)^{2}}}.} Since
575-478: A b u ( t ) d t + i ∫ a b v ( t ) d t . {\displaystyle {\begin{aligned}\int _{a}^{b}f(t)\,dt&=\int _{a}^{b}{\big (}u(t)+iv(t){\big )}\,dt\\&=\int _{a}^{b}u(t)\,dt+i\int _{a}^{b}v(t)\,dt.\end{aligned}}} Now, to define the contour integral, let f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } be
690-2603: A 1 1 + a 2 z 2 + a 3 1 + ⋯ , a m = m 2 . {\displaystyle \operatorname {erfc} z={\frac {z}{\sqrt {\pi }}}e^{-z^{2}}{\cfrac {1}{z^{2}+{\cfrac {a_{1}}{1+{\cfrac {a_{2}}{z^{2}+{\cfrac {a_{3}}{1+\dotsb }}}}}}}},\qquad a_{m}={\frac {m}{2}}.} The inverse factorial series : erfc z = e − z 2 π z ∑ n = 0 ∞ ( − 1 ) n Q n ( z 2 + 1 ) n ¯ = e − z 2 π z [ 1 − 1 2 1 ( z 2 + 1 ) + 1 4 1 ( z 2 + 1 ) ( z 2 + 2 ) − ⋯ ] {\displaystyle {\begin{aligned}\operatorname {erfc} z&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}Q_{n}}{{\left(z^{2}+1\right)}^{\bar {n}}}}\\[1ex]&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\left[1-{\frac {1}{2}}{\frac {1}{(z^{2}+1)}}+{\frac {1}{4}}{\frac {1}{\left(z^{2}+1\right)\left(z^{2}+2\right)}}-\cdots \right]\end{aligned}}} converges for Re( z ) > 0 . Here Q n = def 1 Γ ( 1 2 ) ∫ 0 ∞ τ ( τ − 1 ) ⋯ ( τ − n + 1 ) τ − 1 2 e − τ d τ = ∑ k = 0 n ( 1 2 ) k ¯ s ( n , k ) , {\displaystyle {\begin{aligned}Q_{n}&{\overset {\text{def}}{{}={}}}{\frac {1}{\Gamma {\left({\frac {1}{2}}\right)}}}\int _{0}^{\infty }\tau (\tau -1)\cdots (\tau -n+1)\tau ^{-{\frac {1}{2}}}e^{-\tau }\,d\tau \\[1ex]&=\sum _{k=0}^{n}\left({\frac {1}{2}}\right)^{\bar {k}}s(n,k),\end{aligned}}} z denotes
805-584: A π ( a 2 − 1 ) 2 → 0 as a → ∞ . {\displaystyle \left|\int _{\text{Arc}}f(z)\,dz\right|\leq {\frac {a\pi }{\left(a^{2}-1\right)^{2}}}\to 0{\text{ as }}a\to \infty .} So ∫ − ∞ ∞ 1 ( x 2 + 1 ) 2 d x = ∫ − ∞ ∞ f ( z ) d z = lim
920-1271: A → ∞ . {\displaystyle \int _{\text{arc}}{\frac {e^{itz}}{z^{2}+1}}\,dz\rightarrow 0{\mbox{ as }}a\rightarrow \infty .} Therefore, if t > 0 then ∫ − ∞ ∞ e i t x x 2 + 1 d x = π e − t . {\displaystyle \int _{-\infty }^{\infty }{\frac {e^{itx}}{x^{2}+1}}\,dx=\pi e^{-t}.} A similar argument with an arc that winds around − i rather than i shows that if t < 0 then ∫ − ∞ ∞ e i t x x 2 + 1 d x = π e t , {\displaystyle \int _{-\infty }^{\infty }{\frac {e^{itx}}{x^{2}+1}}\,dx=\pi e^{t},} and finally we have this: ∫ − ∞ ∞ e i t x x 2 + 1 d x = π e − | t | . {\displaystyle \int _{-\infty }^{\infty }{\frac {e^{itx}}{x^{2}+1}}\,dx=\pi e^{-|t|}.} (If t = 0 then
1035-489: A → + ∞ ∫ − a a f ( z ) d z = π 2 . ◻ {\displaystyle \int _{-\infty }^{\infty }{\frac {1}{\left(x^{2}+1\right)^{2}}}\,dx=\int _{-\infty }^{\infty }f(z)\,dz=\lim _{a\to +\infty }\int _{-a}^{a}f(z)\,dz={\frac {\pi }{2}}.\quad \square } Consider the Laurent series of f ( z ) about i ,
1150-473: A + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i , ( a + b i ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i . {\displaystyle {\begin{aligned}(a+bi)+(c+di)&=(a+c)+(b+d)i,\\[5mu](a+bi)(c+di)&=(ac-bd)+(ad+bc)i.\end{aligned}}} When multiplied by
1265-673: A + b i ) = − b + a i , − i ( a + b i ) = b − a i . {\displaystyle i(a+bi)=-b+ai,\quad -i(a+bi)=b-ai.} The powers of i repeat in a cycle expressible with the following pattern, where n is any integer: i 4 n = 1 , i 4 n + 1 = i , i 4 n + 2 = − 1 , i 4 n + 3 = − i . {\displaystyle i^{4n}=1,\quad i^{4n+1}=i,\quad i^{4n+2}=-1,\quad i^{4n+3}=-i.} Thus, under multiplication, i
SECTION 10
#17327931103531380-417: A , b ] → C {\displaystyle z:[a,b]\to \mathbb {C} } . This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration. In the following subsections we narrow down
1495-666: A 1 = 0.0705230784 , a 2 = 0.0422820123 , a 3 = 0.0092705272 , a 4 = 0.0001520143 , a 5 = 0.0002765672 , a 6 = 0.0000430638 erf x ≈ 1 − ( a 1 t + a 2 t 2 + ⋯ + a 5 t 5 ) e − x 2 , t = 1 1 + p x {\displaystyle \operatorname {erf} x\approx 1-\left(a_{1}t+a_{2}t^{2}+\cdots +a_{5}t^{5}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}}} (maximum error: 1.5 × 10 ) where p = 0.3275911 ,
1610-552: A 1 = 0.254829592 , a 2 = −0.284496736 , a 3 = 1.421413741 , a 4 = −1.453152027 , a 5 = 1.061405429 All of these approximations are valid for x ≥ 0 . To use these approximations for negative x , use the fact that erf x is an odd function, so erf x = −erf(− x ) . This approximation can be inverted to obtain an approximation for the inverse error function: erf − 1 x ≈ sgn x ⋅ ( 2 π
1725-613: A 1 = 0.278393 , a 2 = 0.230389 , a 3 = 0.000972 , a 4 = 0.078108 erf x ≈ 1 − ( a 1 t + a 2 t 2 + a 3 t 3 ) e − x 2 , t = 1 1 + p x , x ≥ 0 {\displaystyle \operatorname {erf} x\approx 1-\left(a_{1}t+a_{2}t^{2}+a_{3}t^{3}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}},\qquad x\geq 0} (maximum error: 2.5 × 10 ) where p = 0.47047 ,
1840-524: A 1 = 0.3480242 , a 2 = −0.0958798 , a 3 = 0.7478556 erf x ≈ 1 − 1 ( 1 + a 1 x + a 2 x 2 + ⋯ + a 6 x 6 ) 16 , x ≥ 0 {\displaystyle \operatorname {erf} x\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+\cdots +a_{6}x^{6}\right)^{16}}},\qquad x\geq 0} (maximum error: 3 × 10 ) where
1955-1029: A l l a c y ( − 1 ) ⋅ ( − 1 ) = 1 = 1 (incorrect). {\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}\mathrel {\stackrel {\mathrm {fallacy} }{=}} {\textstyle {\sqrt {(-1)\cdot (-1)}}}={\sqrt {1}}=1\qquad {\text{(incorrect).}}} Generally, the calculation rules x t y ⋅ y t y = x ⋅ y t y {\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}} and x t y / y t y = x / y {\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}} are guaranteed to be valid only for real, positive values of x and y . When x or y
2070-478: A + bi can be represented by: a I + b J = ( a − b b a ) . {\displaystyle aI+bJ={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.} More generally, any real-valued 2 × 2 matrix with a trace of zero and a determinant of one squares to − I , so could be chosen for J . Larger matrices could also be used; for example, 1 could be represented by
2185-405: A complex function f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } is a generalization of the integral for real-valued functions. For continuous functions in the complex plane , the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over
2300-448: A continuous function on the directed smooth curve γ {\displaystyle \gamma } . Let z : [ a , b ] → C {\displaystyle z:[a,b]\to \mathbb {C} } be any parametrization of γ {\displaystyle \gamma } that is consistent with its order (direction). Then the integral along γ {\displaystyle \gamma }
2415-480: A number line , the imaginary axis , which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally. Integer sums of the real unit 1 and the imaginary unit i form a square lattice in the complex plane called the Gaussian integers . The sum, difference, or product of Gaussian integers is also a Gaussian integer: (
SECTION 20
#17327931103532530-703: A "straight" part and a curved arc, so that ∫ straight + ∫ arc = π e − t , {\displaystyle \int _{\text{straight}}+\int _{\text{arc}}=\pi e^{-t},} and thus ∫ − a a = π e − t − ∫ arc . {\displaystyle \int _{-a}^{a}=\pi e^{-t}-\int _{\text{arc}}.} According to Jordan's lemma , if t > 0 then ∫ arc e i t z z 2 + 1 d z → 0 as
2645-569: A ) will be convenient. Call this contour C . There are two ways of proceeding, using the Cauchy integral formula or by the method of residues: Note that: ∮ C f ( z ) d z = ∫ − a a f ( z ) d z + ∫ Arc f ( z ) d z {\displaystyle \oint _{C}f(z)\,dz=\int _{-a}^{a}f(z)\,dz+\int _{\text{Arc}}f(z)\,dz} thus ∫ −
2760-472: A complex number z , there is not a unique complex number w satisfying erf w = z , so a true inverse function would be multivalued. However, for −1 < x < 1 , there is a unique real number denoted erf x satisfying erf ( erf − 1 x ) = x . {\displaystyle \operatorname {erf} \left(\operatorname {erf} ^{-1}x\right)=x.} The inverse error function
2875-738: A contour, parametrized by z ( t ) = e , with t ∈ [0, 2π] , then dz / dt = ie and ∮ C 1 z d z = ∫ 0 2 π 1 e i t i e i t d t = i ∫ 0 2 π 1 d t = i t | 0 2 π = ( 2 π − 0 ) i = 2 π i {\displaystyle \oint _{C}{\frac {1}{z}}\,dz=\int _{0}^{2\pi }{\frac {1}{e^{it}}}ie^{it}\,dt=i\int _{0}^{2\pi }1\,dt=i\,t{\Big |}_{0}^{2\pi }=\left(2\pi -0\right)i=2\pi i} which
2990-402: A non-vanishing, continuous derivative such that each point is traversed only once ( z is one-to-one), with the possible exception of a curve such that the endpoints match ( z ( a ) = z ( b ) {\displaystyle z(a)=z(b)} ). In the case where the endpoints match, the curve is called closed, and the function is required to be one-to-one everywhere else and
3105-445: A pen, in a sequence of even, steady strokes, which stop only to start a new piece of the curve, all without picking up the pen. Contours are often defined in terms of directed smooth curves. These provide a precise definition of a "piece" of a smooth curve, of which a contour is made. A smooth curve is a curve z : [ a , b ] → C {\displaystyle z:[a,b]\to \mathbb {C} } with
3220-457: A real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval and the Riemann integral . In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour. To define the contour integral in this way one must first consider the integral, over
3335-651: A real variable, of a complex-valued function. Let f : R → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } be a complex-valued function of a real variable, t {\displaystyle t} . The real and imaginary parts of f {\displaystyle f} are often denoted as u ( t ) {\displaystyle u(t)} and v ( t ) {\displaystyle v(t)} , respectively, so that f ( t ) = u ( t ) + i v ( t ) . {\displaystyle f(t)=u(t)+iv(t).} Then
3450-414: A representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is isomorphic to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects. More generally, in the geometric algebra of any higher-dimensional Euclidean space ,
3565-421: A right angle between them. Addition by a complex number corresponds to translation in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every similarity transformation of the plane can be represented by a complex-linear function z ↦ a z + b . {\displaystyle z\mapsto az+b.} In the geometric algebra of
Error function - Misplaced Pages Continue
3680-413: A scalar multiple of the characteristic function of the Cauchy distribution ) resists the techniques of elementary calculus . We will evaluate it by expressing it as a limit of contour integrals along the contour C that goes along the real line from − a to a and then counterclockwise along a semicircle centered at 0 from a to − a . Take a to be greater than 1, so that the imaginary unit i
3795-645: A single direction. This requires that the sequence of curves γ 1 , … , γ n {\displaystyle \gamma _{1},\dots ,\gamma _{n}} be such that the terminal point of γ i {\displaystyle \gamma _{i}} coincides with the initial point of γ i + 1 {\displaystyle \gamma _{i+1}} for all i {\displaystyle i} such that 1 ≤ i < n {\displaystyle 1\leq i<n} . This includes all directed smooth curves. Also,
3910-606: A single point in the complex plane is considered a contour. The symbol + {\displaystyle +} is often used to denote the piecing of curves together to form a new curve. Thus we could write a contour Γ {\displaystyle \Gamma } that is made up of n {\displaystyle n} curves as Γ = γ 1 + γ 2 + ⋯ + γ n . {\displaystyle \Gamma =\gamma _{1}+\gamma _{2}+\cdots +\gamma _{n}.} The contour integral of
4025-449: A unit bivector of any arbitrary planar orientation squares to −1 , so can be taken to represent the imaginary unit i . The imaginary unit was historically written − 1 , {\textstyle {\sqrt {-1}},} and still is in some modern works. However, great care needs to be taken when manipulating formulas involving radicals . The radical sign notation x {\textstyle {\sqrt {x}}}
4140-430: Is isomorphic to the complex numbers, and the variable x {\displaystyle x} expresses the imaginary unit. The complex numbers can be represented graphically by drawing the real number line as the horizontal axis and the imaginary numbers as the vertical axis of a Cartesian plane called the complex plane . In this representation, the numbers 1 and i are at the same distance from 0 , with
4255-432: Is normally distributed with mean 0 and standard deviation 1 / 2 {\displaystyle 1/{\sqrt {2}}} , erf x is the probability that Y falls in the range [− x , x ] . Two closely related functions are the complementary error function e r f c : C → C {\displaystyle \mathrm {erfc} :\mathbb {C} \to \mathbb {C} }
4370-414: Is a complex contour integral which is path-independent because exp ( − t 2 ) {\displaystyle \exp(-t^{2})} is holomorphic on the whole complex plane C {\displaystyle \mathbb {C} } . In many applications, the function argument is a real number, in which case the function value is also real. In some old texts,
4485-837: Is a generator of a cyclic group of order 4, a discrete subgroup of the continuous circle group of the unit complex numbers under multiplication. Written as a special case of Euler's formula for an integer n , i n = exp ( 1 2 π i ) n = exp ( 1 2 n π i ) = cos ( 1 2 n π ) + i sin ( 1 2 n π ) . {\displaystyle i^{n}={\exp }{\bigl (}{\tfrac {1}{2}}\pi i{\bigr )}^{n}={\exp }{\bigl (}{\tfrac {1}{2}}n\pi i{\bigr )}={\cos }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}+{i\sin }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}.} With
4600-425: Is a negative scalar. The quotient of a vector with itself is the scalar 1 = u / u , and when multiplied by any vector leaves it unchanged (the identity transformation ). The quotient of any two perpendicular vectors of the same magnitude, J = u / v , which when multiplied rotates the divisor a quarter turn into the dividend, Jv = u , is a unit bivector which squares to −1 , and can thus be taken as
4715-461: Is a solution to the quadratic equation x + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers , using addition and multiplication . A simple example of the use of i in a complex number is 2 + 3 i . Imaginary numbers are an important mathematical concept; they extend the real number system R {\displaystyle \mathbb {R} } to
Error function - Misplaced Pages Continue
4830-747: Is a unique real number erfi x satisfying erfi(erfi x ) = x . The inverse imaginary error function is defined as erfi x . For any real x , Newton's method can be used to compute erfi x , and for −1 ≤ x ≤ 1 , the following Maclaurin series converges: erfi − 1 z = ∑ k = 0 ∞ ( − 1 ) k c k 2 k + 1 ( π 2 z ) 2 k + 1 , {\displaystyle \operatorname {erfi} ^{-1}z=\sum _{k=0}^{\infty }{\frac {(-1)^{k}c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},} where c k
4945-638: Is defined as erfc x = 1 − erf x = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx x , {\displaystyle {\begin{aligned}\operatorname {erfc} x&=1-\operatorname {erf} x\\[5pt]&={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t\\[5pt]&=e^{-x^{2}}\operatorname {erfcx} x,\end{aligned}}} which also defines erfcx ,
5060-561: Is defined as erfc z = 1 − erf z , {\displaystyle \operatorname {erfc} z=1-\operatorname {erf} z,} and the imaginary error function e r f i : C → C {\displaystyle \mathrm {erfi} :\mathbb {C} \to \mathbb {C} } is defined as erfi z = − i erf i z , {\displaystyle \operatorname {erfi} z=-i\operatorname {erf} iz,} where i
5175-431: Is defined as Contour integration Contour integration is closely related to the calculus of residues , a method of complex analysis . One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: One method can be used, or a combination of these methods, or various limiting processes, for
5290-1270: Is defined as above. A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real x is erfc x = e − x 2 x π ( 1 + ∑ n = 1 ∞ ( − 1 ) n 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) ( 2 x 2 ) n ) = e − x 2 x π ∑ n = 0 ∞ ( − 1 ) n ( 2 n − 1 ) ! ! ( 2 x 2 ) n , {\displaystyle {\begin{aligned}\operatorname {erfc} x&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left(1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {1\cdot 3\cdot 5\cdots (2n-1)}{\left(2x^{2}\right)^{n}}}\right)\\[6pt]&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{\left(2x^{2}\right)^{n}}},\end{aligned}}} where (2 n − 1)!!
5405-496: Is denoted ∫ γ f ( z ) d z {\displaystyle \int _{\gamma }f(z)\,dz\,} and is given by ∫ γ f ( z ) d z := ∫ a b f ( z ( t ) ) z ′ ( t ) d t . {\displaystyle \int _{\gamma }f(z)\,dz:=\int _{a}^{b}f{\big (}z(t){\big )}z'(t)\,dt.} This definition
5520-414: Is enclosed within the curve. The contour integral is ∫ C e i t z z 2 + 1 d z . {\displaystyle \int _{C}{\frac {e^{itz}}{z^{2}+1}}\,dz.} Since e is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator z + 1
5635-482: Is inherently positive or negative in the sense that real numbers are. A more formal expression of this indistinguishability of + i and − i is that, although the complex field is unique (as an extension of the real numbers) up to isomorphism , it is not unique up to a unique isomorphism. That is, there are two field automorphisms of the complex numbers C {\displaystyle \mathbb {C} } that keep each real number fixed, namely
5750-434: Is real but negative, these problems can be avoided by writing and manipulating expressions like i 7 {\textstyle i{\sqrt {7}}} , rather than − 7 {\textstyle {\sqrt {-7}}} . For a more thorough discussion, see the articles Square root and Branch point . As a complex number, the imaginary unit follows all of the rules of complex arithmetic . When
5865-500: Is reserved either for the principal square root function, which is defined for only real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results: − 1 = i ⋅ i = − 1 ⋅ − 1 = f
SECTION 50
#17327931103535980-460: Is said to have an argument of − π 2 , {\displaystyle -{\tfrac {\pi }{2}},} related to the convention of labelling orientations in the Cartesian plane relative to the positive x -axis with positive angles turning anticlockwise in the direction of the positive y -axis. Also, despite the signs written with them, neither + i nor − i
6095-432: Is sufficiently far from the mean, specifically μ − L ≥ σ √ ln k , then: Pr [ X ≤ L ] ≤ A exp ( − B ln k ) = A k B {\displaystyle \Pr[X\leq L]\leq A\exp(-B\ln {k})={\frac {A}{k^{B}}}} so the probability goes to 0 as k → ∞ . The probability for X being in
6210-573: Is the complex conjugate of z . The integrand f = exp(− z ) and f = erf z are shown in the complex z -plane in the figures at right with domain coloring . The error function at +∞ is exactly 1 (see Gaussian integral ). At the real axis, erf z approaches unity at z → +∞ and −1 at z → −∞ . At the imaginary axis, it tends to ± i ∞ . The error function is an entire function ; it has no singularities (except that at infinity) and its Taylor expansion always converges. For x >> 1 , however, cancellation of leading terms makes
6325-778: Is the double factorial of (2 n − 1) , which is the product of all odd numbers up to (2 n − 1) . This series diverges for every finite x , and its meaning as asymptotic expansion is that for any integer N ≥ 1 one has erfc x = e − x 2 x π ∑ n = 0 N − 1 ( − 1 ) n ( 2 n − 1 ) ! ! ( 2 x 2 ) n + R N ( x ) {\displaystyle \operatorname {erfc} x={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{N-1}(-1)^{n}{\frac {(2n-1)!!}{\left(2x^{2}\right)^{n}}}+R_{N}(x)} where
6440-673: Is the imaginary unit . The name "error function" and its abbreviation erf were proposed by J. W. L. Glaisher in 1871 on account of its connection with "the theory of Probability, and notably the theory of Errors ." The error function complement was also discussed by Glaisher in a separate publication in the same year. For the "law of facility" of errors whose density is given by f ( x ) = ( c π ) 1 / 2 e − c x 2 {\displaystyle f(x)=\left({\frac {c}{\pi }}\right)^{1/2}e^{-cx^{2}}} (the normal distribution ), Glaisher calculates
6555-923: Is the sign function . By keeping only the first two coefficients and choosing c 1 = 31 / 200 and c 2 = − 341 / 8000 , the resulting approximation shows its largest relative error at x = ±1.3796 , where it is less than 0.0036127: erf x ≈ 2 π sgn x ⋅ 1 − e − x 2 ( π 2 + 31 200 e − x 2 − 341 8000 e − 2 x 2 ) . {\displaystyle \operatorname {erf} x\approx {\frac {2}{\sqrt {\pi }}}\operatorname {sgn} x\cdot {\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+{\frac {31}{200}}e^{-x^{2}}-{\frac {341}{8000}}e^{-2x^{2}}\right).} Given
6670-423: Is the value of the integral. This result only applies to the case in which z is raised to power of -1. If the power is not equal to -1, then the result will always be zero. Applications of integral theorems are also often used to evaluate the contour integral along a contour, which means that the real-valued integral is calculated simultaneously along with calculating the contour integral. Integral theorems such as
6785-1410: Is usually defined with domain (−1,1) , and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk | z | < 1 of the complex plane, using the Maclaurin series erf − 1 z = ∑ k = 0 ∞ c k 2 k + 1 ( π 2 z ) 2 k + 1 , {\displaystyle \operatorname {erf} ^{-1}z=\sum _{k=0}^{\infty }{\frac {c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},} where c 0 = 1 and c k = ∑ m = 0 k − 1 c m c k − 1 − m ( m + 1 ) ( 2 m + 1 ) = { 1 , 1 , 7 6 , 127 90 , 4369 2520 , 34807 16200 , … } . {\displaystyle {\begin{aligned}c_{k}&=\sum _{m=0}^{k-1}{\frac {c_{m}c_{k-1-m}}{(m+1)(2m+1)}}\\[1ex]&=\left\{1,1,{\frac {7}{6}},{\frac {127}{90}},{\frac {4369}{2520}},{\frac {34807}{16200}},\ldots \right\}.\end{aligned}}} So we have
6900-1036: Is valid only for positive values of x , but it can be used in conjunction with erfc x = 2 − erfc(− x ) to obtain erfc( x ) for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the erfc of the sum of two non-negative variables is as follows: erfc ( x + y ∣ x , y ≥ 0 ) = 2 π ∫ 0 π 2 exp ( − x 2 sin 2 θ − y 2 cos 2 θ ) d θ . {\displaystyle \operatorname {erfc} (x+y\mid x,y\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}-{\frac {y^{2}}{\cos ^{2}\theta }}\right)\,\mathrm {d} \theta .} The imaginary error function , denoted erfi ,
7015-400: Is well defined. That is, the result is independent of the parametrization chosen. In the case where the real integral on the right side does not exist the integral along γ {\displaystyle \gamma } is said not to exist. The generalization of the Riemann integral to functions of a complex variable is done in complete analogy to its definition for functions from
SECTION 60
#17327931103537130-1018: Is zero. Since z + 1 = ( z + i )( z − i ) , that happens only where z = i or z = − i . Only one of those points is in the region bounded by this contour. The residue of f ( z ) at z = i is lim z → i ( z − i ) f ( z ) = lim z → i ( z − i ) e i t z z 2 + 1 = lim z → i ( z − i ) e i t z ( z − i ) ( z + i ) = lim z → i e i t z z + i = e − t 2 i . {\displaystyle \lim _{z\to i}(z-i)f(z)=\lim _{z\to i}(z-i){\frac {e^{itz}}{z^{2}+1}}=\lim _{z\to i}(z-i){\frac {e^{itz}}{(z-i)(z+i)}}=\lim _{z\to i}{\frac {e^{itz}}{z+i}}={\frac {e^{-t}}{2i}}.} According to
7245-510: The 4 × 4 identity matrix and i could be represented by any of the Dirac matrices for spatial dimensions. Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose coefficients are real numbers form a ring , denoted R [ x ] , {\displaystyle \mathbb {R} [x],} an algebraic structure with addition and multiplication and sharing many properties with
7360-418: The Cauchy integral formula or residue theorem are generally used in the following method: Consider the integral ∫ − ∞ ∞ 1 ( x 2 + 1 ) 2 d x , {\displaystyle \int _{-\infty }^{\infty }{\frac {1}{\left(x^{2}+1\right)^{2}}}\,dx,} To evaluate this integral, we look at
7475-458: The Euclidean plane , the geometric product or quotient of two arbitrary vectors is a sum of a scalar (real number) part and a bivector part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector
7590-1107: The OEIS . For iterative calculation of the above series, the following alternative formulation may be useful: erf z = 2 π ∑ n = 0 ∞ ( z ∏ k = 1 n − ( 2 k − 1 ) z 2 k ( 2 k + 1 ) ) = 2 π ∑ n = 0 ∞ z 2 n + 1 ∏ k = 1 n − z 2 k {\displaystyle {\begin{aligned}\operatorname {erf} z&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }\left(z\prod _{k=1}^{n}{\frac {-(2k-1)z^{2}}{k(2k+1)}}\right)\\[6pt]&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z}{2n+1}}\prod _{k=1}^{n}{\frac {-z^{2}}{k}}\end{aligned}}} because −(2 k − 1) z / k (2 k + 1) expresses
7705-433: The complex plane , which is a special interpretation of a Cartesian plane , i is the point located one unit from the origin along the imaginary axis (which is orthogonal to the real axis ). Being a quadratic polynomial with no multiple root , the defining equation x = −1 has two distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. Although
7820-1152: The heat equation when boundary conditions are given by the Heaviside step function . The error function and its approximations can be used to estimate results that hold with high probability or with low probability. Given a random variable X ~ Norm[ μ , σ ] (a normal distribution with mean μ and standard deviation σ ) and a constant L > μ , it can be shown via integration by substitution: Pr [ X ≤ L ] = 1 2 + 1 2 erf L − μ 2 σ ≈ A exp ( − B ( L − μ σ ) 2 ) {\displaystyle {\begin{aligned}\Pr[X\leq L]&={\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} {\frac {L-\mu }{{\sqrt {2}}\sigma }}\\&\approx A\exp \left(-B\left({\frac {L-\mu }{\sigma }}\right)^{2}\right)\end{aligned}}} where A and B are certain numeric constants. If L
7935-488: The residue theorem , then, we have ∫ C f ( z ) d z = 2 π i Res z = i f ( z ) = 2 π i e − t 2 i = π e − t . {\displaystyle \int _{C}f(z)\,dz=2\pi i\operatorname {Res} _{z=i}f(z)=2\pi i{\frac {e^{-t}}{2i}}=\pi e^{-t}.} The contour C may be split into
8050-629: The rising factorial , and s ( n , k ) denotes a signed Stirling number of the first kind . There also exists a representation by an infinite sum containing the double factorial : erf z = 2 π ∑ n = 0 ∞ ( − 2 ) n ( 2 n − 1 ) ! ! ( 2 n + 1 ) ! z 2 n + 1 {\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-2)^{n}(2n-1)!!}{(2n+1)!}}z^{2n+1}} where
8165-725: The scaled complementary error function (which can be used instead of erfc to avoid arithmetic underflow ). Another form of erfc x for x ≥ 0 is known as Craig's formula, after its discoverer: erfc ( x ∣ x ≥ 0 ) = 2 π ∫ 0 π 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname {erfc} (x\mid x\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}\right)\,\mathrm {d} \theta .} This expression
8280-1306: The Taylor expansion unpractical. The defining integral cannot be evaluated in closed form in terms of elementary functions (see Liouville's theorem ), but by expanding the integrand e into its Maclaurin series and integrating term by term, one obtains the error function's Maclaurin series as: erf z = 2 π ∑ n = 0 ∞ ( − 1 ) n z 2 n + 1 n ! ( 2 n + 1 ) = 2 π ( z − z 3 3 + z 5 10 − z 7 42 + z 9 216 − ⋯ ) {\displaystyle {\begin{aligned}\operatorname {erf} z&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{n!(2n+1)}}\\[6pt]&={\frac {2}{\sqrt {\pi }}}\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{10}}-{\frac {z^{7}}{42}}+{\frac {z^{9}}{216}}-\cdots \right)\end{aligned}}} which holds for every complex number z . The denominator terms are sequence A007680 in
8395-538: The complex number system C , {\displaystyle \mathbb {C} ,} in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra ). Here, the term "imaginary" is used because there is no real number having a negative square . There are two complex square roots of −1: i and − i , just as there are two complex square roots of every real number other than zero (which has one double square root ). In contexts in which use of
8510-402: The complex-valued function f ( z ) = 1 ( z 2 + 1 ) 2 {\displaystyle f(z)={\frac {1}{\left(z^{2}+1\right)^{2}}}} which has singularities at i and − i . We choose a contour that will enclose the real-valued integral, here a semicircle with boundary diameter on the real line (going from, say, − a to
8625-1045: The construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of i with −1 ). Higher integral powers of i are thus i 3 = i 2 i = ( − 1 ) i = − i , i 4 = i 3 i = ( − i ) i = 1 , i 5 = i 4 i = ( 1 ) i = i , {\displaystyle {\begin{alignedat}{3}i^{3}&=i^{2}i&&=(-1)i&&=-i,\\[3mu]i^{4}&=i^{3}i&&=\;\!(-i)i&&=\ \,1,\\[3mu]i^{5}&=i^{4}i&&=\ \,(1)i&&=\ \ i,\end{alignedat}}} and so on, cycling through
8740-571: The derivative must be continuous at the identified point ( z ′ ( a ) = z ′ ( b ) {\displaystyle z'(a)=z'(b)} ). A smooth curve that is not closed is often referred to as a smooth arc. The parametrization of a curve provides a natural ordering of points on the curve: z ( x ) {\displaystyle z(x)} comes before z ( y ) {\displaystyle z(y)} if x < y {\displaystyle x<y} . This leads to
8855-673: The error function follows immediately from its definition: d d z erf z = 2 π e − z 2 . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} z={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}.} From this, the derivative of the imaginary error function is also immediate: d d z erfi z = 2 π e z 2 . {\displaystyle {\frac {d}{dz}}\operatorname {erfi} z={\frac {2}{\sqrt {\pi }}}e^{z^{2}}.} An antiderivative of
8970-571: The error function is an odd function . This directly results from the fact that the integrand e is an even function (the antiderivative of an even function which is zero at the origin is an odd function and vice versa). Since the error function is an entire function which takes real numbers to real numbers, for any complex number z : erf z ¯ = erf z ¯ {\displaystyle \operatorname {erf} {\overline {z}}={\overline {\operatorname {erf} z}}} where z
9085-431: The error function is defined without the factor of 2 / π {\displaystyle 2/{\sqrt {\pi }}} . This nonelementary integral is a sigmoid function that occurs often in probability , statistics , and partial differential equations . In statistics, for non-negative real values of x , the error function has the following interpretation: for a real random variable Y that
9200-1356: The error function, obtainable by integration by parts , is z erf z + e − z 2 π . {\displaystyle z\operatorname {erf} z+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}.} An antiderivative of the imaginary error function, also obtainable by integration by parts, is z erfi z − e z 2 π . {\displaystyle z\operatorname {erfi} z-{\frac {e^{z^{2}}}{\sqrt {\pi }}}.} Higher order derivatives are given by erf ( k ) z = 2 ( − 1 ) k − 1 π H k − 1 ( z ) e − z 2 = 2 π d k − 1 d z k − 1 ( e − z 2 ) , k = 1 , 2 , … {\displaystyle \operatorname {erf} ^{(k)}z={\frac {2(-1)^{k-1}}{\sqrt {\pi }}}{\mathit {H}}_{k-1}(z)e^{-z^{2}}={\frac {2}{\sqrt {\pi }}}{\frac {\mathrm {d} ^{k-1}}{\mathrm {d} z^{k-1}}}\left(e^{-z^{2}}\right),\qquad k=1,2,\dots } where H are
9315-426: The first derivative, in the above steps, because the pole is a second-order pole. That is, ( z − i ) is taken to the second power, so we employ the first derivative of f ( z ) . If it were ( z − i ) taken to the third power, we would use the second derivative and divide by 2! , etc. The case of ( z − i ) to the first power corresponds to a zero order derivative—just f ( z ) itself. We need to show that
9430-480: The first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc x (while for not too large values of x , the above Taylor expansion at 0 provides a very fast convergence). A continued fraction expansion of the complementary error function was found by Laplace : erfc z = z π e − z 2 1 z 2 +
9545-548: The four values 1 , i , −1 , and − i . As with any non-zero real number, i = 1. As a complex number, i can be represented in rectangular form as 0 + 1 i , with a zero real component and a unit imaginary component. In polar form , i can be represented as 1 × e (or just e ), with an absolute value (or magnitude) of 1 and an argument (or angle) of π 2 {\displaystyle {\tfrac {\pi }{2}}} radians . (Adding any integer multiple of 2 π to this angle works as well.) In
9660-431: The identity and complex conjugation . For more on this general phenomenon, see Galois group . Using the concepts of matrices and matrix multiplication , complex numbers can be represented in linear algebra. The real unit 1 and imaginary unit i can be represented by any pair of matrices I and J satisfying I = I , IJ = JI = J , and J = − I . Then a complex number a + bi can be represented by
9775-806: The imaginary unit i , any arbitrary complex number in the complex plane is rotated by a quarter turn ( 1 2 π {\displaystyle {\tfrac {1}{2}}\pi } radians or 90° ) anticlockwise . When multiplied by − i , any arbitrary complex number is rotated by a quarter turn clockwise. In polar form: i r e φ i = r e ( φ + π / 2 ) i , − i r e φ i = r e ( φ − π / 2 ) i . {\displaystyle i\,re^{\varphi i}=re^{(\varphi +\pi /2)i},\quad -i\,re^{\varphi i}=re^{(\varphi -\pi /2)i}.} In rectangular form, i (
9890-580: The imaginary unit is repeatedly added or subtracted, the result is some integer times the imaginary unit, an imaginary integer ; any such numbers can be added and the result is also an imaginary integer: a i + b i = ( a + b ) i . {\displaystyle ai+bi=(a+b)i.} Thus, the imaginary unit is the generator of a group under addition, specifically an infinite cyclic group . The imaginary unit can also be multiplied by any arbitrary real number to form an imaginary number . These numbers can be pictured on
10005-441: The integral of the complex-valued function f {\displaystyle f} over the interval [ a , b ] {\displaystyle [a,b]} is given by ∫ a b f ( t ) d t = ∫ a b ( u ( t ) + i v ( t ) ) d t = ∫
10120-491: The integral over the arc of the semicircle tends to zero as a → ∞ , using the estimation lemma | ∫ Arc f ( z ) d z | ≤ M L {\displaystyle \left|\int _{\text{Arc}}f(z)\,dz\right|\leq ML} where M is an upper bound on | f ( z ) | along the arc and L the length of the arc. Now, | ∫ Arc f ( z ) d z | ≤
10235-688: The integral yields immediately to real-valued calculus methods and its value is π .) Certain substitutions can be made to integrals involving trigonometric functions , so the integral is transformed into a rational function of a complex variable and then the above methods can be used in order to evaluate the integral. As an example, consider ∫ − π π 1 1 + 3 ( cos t ) 2 d t . {\displaystyle \int _{-\pi }^{\pi }{\frac {1}{1+3(\cos t)^{2}}}\,dt.} Imaginary unit The imaginary unit or unit imaginary number ( i )
10350-655: The interval [ L a , L b ] can be derived as Pr [ L a ≤ X ≤ L b ] = ∫ L a L b 1 2 π σ exp ( − ( x − μ ) 2 2 σ 2 ) d x = 1 2 ( erf L b − μ 2 σ − erf L
10465-575: The letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in electrical engineering and control systems engineering , the imaginary unit is normally denoted by j instead of i , because i is commonly used to denote electric current . Square roots of negative numbers are called imaginary because in early-modern mathematics , only what are now called real numbers , obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so
10580-610: The matrix aI + bJ , and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic. The most common choice is to represent 1 and i by the 2 × 2 identity matrix I and the matrix J , I = ( 1 0 0 1 ) , J = ( 0 − 1 1 0 ) . {\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.} Then an arbitrary complex number
10695-1031: The multiplier to turn the k th term into the ( k + 1) th term (considering z as the first term). The imaginary error function has a very similar Maclaurin series, which is: erfi z = 2 π ∑ n = 0 ∞ z 2 n + 1 n ! ( 2 n + 1 ) = 2 π ( z + z 3 3 + z 5 10 + z 7 42 + z 9 216 + ⋯ ) {\displaystyle {\begin{aligned}\operatorname {erfi} z&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{n!(2n+1)}}\\[6pt]&={\frac {2}{\sqrt {\pi }}}\left(z+{\frac {z^{3}}{3}}+{\frac {z^{5}}{10}}+{\frac {z^{7}}{42}}+{\frac {z^{9}}{216}}+\cdots \right)\end{aligned}}} which holds for every complex number z . The derivative of
10810-442: The notion of a directed smooth curve . It is most useful to consider curves independent of the specific parametrization. This can be done by considering equivalence classes of smooth curves with the same direction. A directed smooth curve can then be defined as an ordered set of points in the complex plane that is the image of some smooth curve in their natural order (according to the parametrization). Note that not all orderings of
10925-683: The numerator and denominator values in OEIS : A092676 and OEIS : A092677 respectively; without cancellation the numerator terms are values in OEIS : A002067 .) The error function's value at ±∞ is equal to ±1 . For | z | < 1 , we have erf(erf z ) = z . The inverse complementary error function is defined as erfc − 1 ( 1 − z ) = erf − 1 z . {\displaystyle \operatorname {erfc} ^{-1}(1-z)=\operatorname {erf} ^{-1}z.} For real x , there
11040-1206: The only singularity in the contour is the one at i , then we can write f ( z ) = 1 ( z + i ) 2 ( z − i ) 2 , {\displaystyle f(z)={\frac {\frac {1}{(z+i)^{2}}}{(z-i)^{2}}},} which puts the function in the form for direct application of the formula. Then, by using Cauchy's integral formula, ∮ C f ( z ) d z = ∮ C 1 ( z + i ) 2 ( z − i ) 2 d z = 2 π i d d z 1 ( z + i ) 2 | z = i = 2 π i [ − 2 ( z + i ) 3 ] z = i = π 2 {\displaystyle \oint _{C}f(z)\,dz=\oint _{C}{\frac {\frac {1}{(z+i)^{2}}}{(z-i)^{2}}}\,dz=2\pi i\,\left.{\frac {d}{dz}}{\frac {1}{(z+i)^{2}}}\right|_{z=i}=2\pi i\left[{\frac {-2}{(z+i)^{3}}}\right]_{z=i}={\frac {\pi }{2}}} We take
11155-592: The only singularity we need to consider. We then have f ( z ) = − 1 4 ( z − i ) 2 + − i 4 ( z − i ) + 3 16 + i 8 ( z − i ) + − 5 64 ( z − i ) 2 + ⋯ {\displaystyle f(z)={\frac {-1}{4(z-i)^{2}}}+{\frac {-i}{4(z-i)}}+{\frac {3}{16}}+{\frac {i}{8}}(z-i)+{\frac {-5}{64}}(z-i)^{2}+\cdots } (See
11270-405: The partition (in the two-dimensional complex plane), also known as the mesh, goes to zero. Direct methods involve the calculation of the integral through methods similar to those in calculating line integrals in multivariate calculus. This means that we use the following method: A fundamental result in complex analysis is that the contour integral of 1 / z is 2π i , where
11385-422: The path of the contour is taken to be the unit circle traversed counterclockwise (or any positively oriented Jordan curve about 0). In the case of the unit circle there is a direct method to evaluate the integral ∮ C 1 z d z . {\displaystyle \oint _{C}{\frac {1}{z}}\,dz.} In evaluating this integral, use the unit circle | z | = 1 as
11500-1814: The physicists' Hermite polynomials . An expansion, which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using Hans Heinrich Bürmann 's theorem: erf x = 2 π sgn x ⋅ 1 − e − x 2 ( 1 − 1 12 ( 1 − e − x 2 ) − 7 480 ( 1 − e − x 2 ) 2 − 5 896 ( 1 − e − x 2 ) 3 − 787 276480 ( 1 − e − x 2 ) 4 − ⋯ ) = 2 π sgn x ⋅ 1 − e − x 2 ( π 2 + ∑ k = 1 ∞ c k e − k x 2 ) . {\displaystyle {\begin{aligned}\operatorname {erf} x&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn} x\cdot {\sqrt {1-e^{-x^{2}}}}\left(1-{\frac {1}{12}}\left(1-e^{-x^{2}}\right)-{\frac {7}{480}}\left(1-e^{-x^{2}}\right)^{2}-{\frac {5}{896}}\left(1-e^{-x^{2}}\right)^{3}-{\frac {787}{276480}}\left(1-e^{-x^{2}}\right)^{4}-\cdots \right)\\[10pt]&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn} x\cdot {\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+\sum _{k=1}^{\infty }c_{k}e^{-kx^{2}}\right).\end{aligned}}} where sgn
11615-443: The points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. Also, a single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints. Contours are the class of curves on which we define contour integration. A contour is a directed curve which is made up of a finite sequence of directed smooth curves whose endpoints are matched to give
11730-645: The probability of an error lying between p and q as: ( c π ) 1 2 ∫ p q e − c x 2 d x = 1 2 ( erf ( q c ) − erf ( p c ) ) . {\displaystyle \left({\frac {c}{\pi }}\right)^{\frac {1}{2}}\int _{p}^{q}e^{-cx^{2}}\,\mathrm {d} x={\tfrac {1}{2}}\left(\operatorname {erf} \left(q{\sqrt {c}}\right)-\operatorname {erf} \left(p{\sqrt {c}}\right)\right).} When
11845-408: The property that its square is −1: i 2 = − 1. {\displaystyle i^{2}=-1.} With i defined this way, it follows directly from algebra that i and − i are both square roots of −1. Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number,
11960-408: The purpose of finding these integrals or sums. In complex analysis a contour is a type of curve in the complex plane . In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane: z : [
12075-418: The real numbers. The partition of a directed smooth curve γ {\displaystyle \gamma } is defined as a finite, ordered set of points on γ {\displaystyle \gamma } . The integral over the curve is the limit of finite sums of function values, taken at the points on the partition, in the limit that the maximum distance between any two successive points on
12190-885: The remainder is R N ( x ) := ( − 1 ) N π 2 1 − 2 N ( 2 N ) ! N ! ∫ x ∞ t − 2 N e − t 2 d t , {\displaystyle R_{N}(x):={\frac {(-1)^{N}}{\sqrt {\pi }}}2^{1-2N}{\frac {(2N)!}{N!}}\int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t,} which follows easily by induction, writing e − t 2 = − ( 2 t ) − 1 ( e − t 2 ) ′ {\displaystyle e^{-t^{2}}=-(2t)^{-1}\left(e^{-t^{2}}\right)'} and integrating by parts. The asymptotic behavior of
12305-1415: The remainder term, in Landau notation , is R N ( x ) = O ( x − ( 1 + 2 N ) e − x 2 ) {\displaystyle R_{N}(x)=O\left(x^{-(1+2N)}e^{-x^{2}}\right)} as x → ∞ . This can be found by R N ( x ) ∝ ∫ x ∞ t − 2 N e − t 2 d t = e − x 2 ∫ 0 ∞ ( t + x ) − 2 N e − t 2 − 2 t x d t ≤ e − x 2 ∫ 0 ∞ x − 2 N e − 2 t x d t ∝ x − ( 1 + 2 N ) e − x 2 . {\displaystyle R_{N}(x)\propto \int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t=e^{-x^{2}}\int _{0}^{\infty }(t+x)^{-2N}e^{-t^{2}-2tx}\,\mathrm {d} t\leq e^{-x^{2}}\int _{0}^{\infty }x^{-2N}e^{-2tx}\,\mathrm {d} t\propto x^{-(1+2N)}e^{-x^{2}}.} For large enough values of x , only
12420-482: The results of a series of measurements are described by a normal distribution with standard deviation σ and expected value 0, then erf ( a / σ √ 2 ) is the probability that the error of a single measurement lies between − a and + a , for positive a . This is useful, for example, in determining the bit error rate of a digital communication system. The error and complementary error functions occur, for example, in solutions of
12535-522: The ring of integers . The polynomial x 2 + 1 {\displaystyle x^{2}+1} has no real-number roots , but the set of all real-coefficient polynomials divisible by x 2 + 1 {\displaystyle x^{2}+1} forms an ideal , and so there is a quotient ring R [ x ] / ⟨ x 2 + 1 ⟩ . {\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .} This quotient ring
12650-703: The same result as before. As an aside, a question can arise whether we do not take the semicircle to include the other singularity, enclosing − i . To have the integral along the real axis moving in the correct direction, the contour must travel clockwise, i.e., in a negative direction, reversing the sign of the integral overall. This does not affect the use of the method of residues by series. The integral ∫ − ∞ ∞ e i t x x 2 + 1 d x {\displaystyle \int _{-\infty }^{\infty }{\frac {e^{itx}}{x^{2}+1}}\,dx} (which arises in probability theory as
12765-827: The sample Laurent calculation from Laurent series for the derivation of this series.) It is clear by inspection that the residue is − i / 4 , so, by the residue theorem , we have ∮ C f ( z ) d z = ∮ C 1 ( z 2 + 1 ) 2 d z = 2 π i Res z = i f ( z ) = 2 π i ( − i 4 ) = π 2 ◻ {\displaystyle \oint _{C}f(z)\,dz=\oint _{C}{\frac {1}{\left(z^{2}+1\right)^{2}}}\,dz=2\pi i\,\operatorname {Res} _{z=i}f(z)=2\pi i\left(-{\frac {i}{4}}\right)={\frac {\pi }{2}}\quad \square } Thus we get
12880-886: The series expansion (common factors have been canceled from numerators and denominators): erf − 1 z = π 2 ( z + π 12 z 3 + 7 π 2 480 z 5 + 127 π 3 40320 z 7 + 4369 π 4 5806080 z 9 + 34807 π 5 182476800 z 11 + ⋯ ) . {\displaystyle \operatorname {erf} ^{-1}z={\frac {\sqrt {\pi }}{2}}\left(z+{\frac {\pi }{12}}z^{3}+{\frac {7\pi ^{2}}{480}}z^{5}+{\frac {127\pi ^{3}}{40320}}z^{7}+{\frac {4369\pi ^{4}}{5806080}}z^{9}+{\frac {34807\pi ^{5}}{182476800}}z^{11}+\cdots \right).} (After cancellation
12995-415: The set of curves that we can integrate to include only those that can be built up out of a finite number of continuous curves that can be given a direction. Moreover, we will restrict the "pieces" from crossing over themselves, and we require that each piece have a finite (non-vanishing) continuous derivative. These requirements correspond to requiring that we consider only curves that can be traced, such as by
13110-399: The square root of a negative number was previously considered undefined or nonsensical. The name imaginary is generally credited to René Descartes , and Isaac Newton used the term as early as 1670. The i notation was introduced by Leonhard Euler . A unit is an undivided whole, and unity or the unit number is the number one ( 1 ). The imaginary unit i is defined solely by
13225-530: The two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled + i (or simply i ) and the other is labelled − i , though it is inherently ambiguous which is which. The only differences between + i and − i arise from this labelling. For example, by convention + i is said to have an argument of + π 2 {\displaystyle +{\tfrac {\pi }{2}}} and − i
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