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132-418: Real analysis is distinguished from complex analysis , which deals with the study of complex numbers and their functions. The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set ( R {\displaystyle \mathbb {R} } ), together with two binary operations denoted + and ⋅ , and

264-482: A {\displaystyle a} and b {\displaystyle b} are distinct real numbers, and we exclude the case of I {\displaystyle I} being empty or consisting of only one point, in particular. Definition. If I ⊂ R {\displaystyle I\subset \mathbb {R} } is a non-degenerate interval, we say that f : I → R {\displaystyle f:I\to \mathbb {R} }

396-547: A {\textstyle a} , b {\textstyle b} are real numbers), that are used for generalization of this notion to other domains: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that | a + b | = s ( a + b ) {\displaystyle |a+b|=s(a+b)} where s = ± 1 {\displaystyle s=\pm 1} , with its sign chosen to make

528-479: A 1 ≥ a 2 ≥ a 3 ≥ ⋯ {\displaystyle a_{1}\geq a_{2}\geq a_{3}\geq \cdots } holds, respectively. If either holds, the sequence is said to be monotonic . The monotonicity is strict if the chained inequalities still hold with ≤ {\displaystyle \leq } or ≥ {\displaystyle \geq } replaced by < or >. Given

660-438: A n {\displaystyle a:\mathbb {N} \to \mathbb {R} :n\mapsto a_{n}} . Each a ( n ) = a n {\displaystyle a(n)=a_{n}} is referred to as a term (or, less commonly, an element ) of the sequence. A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or

792-430: A n {\textstyle \lim _{n\to \infty }a_{n}} exists) is said to be convergent ; otherwise it is divergent . ( See the section on limits and convergence for details. ) A real-valued sequence ( a n ) {\displaystyle (a_{n})} is bounded if there exists M ∈ R {\displaystyle M\in \mathbb {R} } such that |

924-508: A n | < M {\displaystyle |a_{n}|<M} for all n ∈ N {\displaystyle n\in \mathbb {N} } . A real-valued sequence ( a n ) {\displaystyle (a_{n})} is monotonically increasing or decreasing if a 1 ≤ a 2 ≤ a 3 ≤ ⋯ {\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq \cdots } or

1056-510: A n ) {\displaystyle (a_{n})} and term a n {\displaystyle a_{n}} by function f {\displaystyle f} and value f ( x ) {\displaystyle f(x)} and natural numbers N {\displaystyle N} and n {\displaystyle n} by real numbers M {\displaystyle M} and x {\displaystyle x} , respectively) yields

1188-514: A n ) {\displaystyle (a_{n})} be a real-valued sequence. We say that ( a n ) {\displaystyle (a_{n})} converges to a {\displaystyle a} if, for any ε > 0 {\displaystyle \varepsilon >0} , there exists a natural number N {\displaystyle N} such that n ≥ N {\displaystyle n\geq N} implies that |

1320-425: A n = a ; {\displaystyle \lim _{n\to \infty }a_{n}=a;} if ( a n ) {\displaystyle (a_{n})} fails to converge, we say that ( a n ) {\displaystyle (a_{n})} diverges . Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (

1452-407: A − a n | < ε {\displaystyle |a-a_{n}|<\varepsilon } . We write this symbolically as a n → a     as     n → ∞ , {\displaystyle a_{n}\to a\ \ {\text{as}}\ \ n\to \infty ,} or as lim n → ∞

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1584-407: A + b | = s ⋅ ( a + b ) = s ⋅ a + s ⋅ b ≤ | a | + | b | {\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|} , as desired. Some additional useful properties are given below. These are either immediate consequences of the definition or implied by

1716-432: A total order denoted ≤ . The operations make the real numbers a field , and, along with the order, an ordered field . The real number system is the unique complete ordered field , in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g.,

1848-524: A vector-valued function from X into R 2 . {\displaystyle \mathbb {R} ^{2}.} Some properties of complex-valued functions (such as continuity ) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as differentiability , are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every differentiable complex function

1980-463: A vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude . In programming languages and computational software packages, the absolute value of x {\textstyle x} is generally represented by abs( x ) , or a similar expression. The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to

2112-500: A convergent subsequence. This particular property is known as subsequential compactness . In R {\displaystyle \mathbb {R} } , a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above. Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general. The most general definition of compactness relies on

2244-424: A family of functions to uniformly converge, sometimes denoted f n ⇉ f {\displaystyle f_{n}\rightrightarrows f} , such a value of N {\displaystyle N} must exist for any ε > 0 {\displaystyle \varepsilon >0} given, no matter how small. Intuitively, we can visualize this situation by imagining that, for

2376-475: A finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including the complex exponential function , complex logarithm functions , and trigonometric functions . Complex functions that are differentiable at every point of an open subset Ω {\displaystyle \Omega } of the complex plane are said to be holomorphic on Ω {\displaystyle \Omega } . In

2508-461: A finite subcover. Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent. As another example, the image of a compact metric space under a continuous map is also compact. A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; such

2640-512: A function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful residue theorem . The remarkable behavior of holomorphic functions near essential singularities is described by Picard's theorem . Functions that have only poles but no essential singularities are called meromorphic . Laurent series are

2772-433: A function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". There are several ways to make this intuition mathematically rigorous. Several definitions of varying levels of generality can be given. In cases where two or more definitions are applicable, they are readily shown to be equivalent to one another, so the most convenient definition can be used to determine whether

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2904-587: A function is uniformly continuous on X {\displaystyle X} , the choice of δ {\displaystyle \delta } needed to fulfill the definition must work for all of X {\displaystyle X} for a given ε {\displaystyle \varepsilon } . In contrast, when a function is continuous at every point p ∈ X {\displaystyle p\in X} (or said to be continuous on X {\displaystyle X} ),

3036-451: A function, and d d x | f ( x ) | = f ( x ) | f ( x ) | f ′ ( x ) {\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)} if another function is inside the absolute value. In the first case, the derivative is always discontinuous at x = 0 {\textstyle x=0} in

3168-430: A general term enclosed in parentheses: ( a n ) = ( a n ) n ∈ N = ( a 1 , a 2 , a 3 , … ) . {\displaystyle (a_{n})=(a_{n})_{n\in \mathbb {N} }=(a_{1},a_{2},a_{3},\dots ).} A sequence that tends to a limit (i.e., lim n → ∞

3300-537: A given ε > 0 {\displaystyle \varepsilon >0} . Definition. Let I ⊂ R {\displaystyle I\subset \mathbb {R} } be an interval on the real line . A function f : I → R {\displaystyle f:I\to \mathbb {R} } is said to be absolutely continuous on I {\displaystyle I} if for every positive number ε {\displaystyle \varepsilon } , there

3432-411: A given function is continuous or not. In the first definition given below, f : I → R {\displaystyle f:I\to \mathbb {R} } is a function defined on a non-degenerate interval I {\displaystyle I} of the set of real numbers as its domain. Some possibilities include I = R {\displaystyle I=\mathbb {R} } ,

3564-737: A large enough N {\displaystyle N} , the functions f N , f N + 1 , f N + 2 , … {\displaystyle f_{N},f_{N+1},f_{N+2},\ldots } are all confined within a 'tube' of width 2 ε {\displaystyle 2\varepsilon } about f {\displaystyle f} (that is, between f − ε {\displaystyle f-\varepsilon } and f + ε {\displaystyle f+\varepsilon } ) for every value in their domain E {\displaystyle E} . The distinction between pointwise and uniform convergence

3696-448: A limit is fundamental to calculus (and mathematical analysis in general) and its formal definition is used in turn to define notions like continuity , derivatives , and integrals . (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.) The concept of limit was informally introduced for functions by Newton and Leibniz , at

3828-721: A more common and less ambiguous notation. For any real number x {\displaystyle x} , the absolute value or modulus of x {\displaystyle x} is denoted by | x | {\displaystyle |x|} , with a vertical bar on each side of the quantity, and is defined as | x | = { x , if  x ≥ 0 − x , if  x < 0. {\displaystyle |x|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}} The absolute value of x {\displaystyle x}

3960-399: A natural number N {\displaystyle N} such that m , n ≥ N {\displaystyle m,n\geq N} implies that | a m − a n | < ε {\displaystyle |a_{m}-a_{n}|<\varepsilon } . It can be shown that a real-valued sequence is Cauchy if and only if it

4092-532: A sequence ( a n ) {\displaystyle (a_{n})} , another sequence ( b k ) {\displaystyle (b_{k})} is a subsequence of ( a n ) {\displaystyle (a_{n})} if b k = a n k {\displaystyle b_{k}=a_{n_{k}}} for all positive integers k {\displaystyle k} and ( n k ) {\displaystyle (n_{k})}

Real analysis - Misplaced Pages Continue

4224-429: A set X  ×  X is called a metric (or a distance function ) on  X , if it satisfies the following four axioms: The definition of absolute value given for real numbers above can be extended to any ordered ring . That is, if  a is an element of an ordered ring  R , then the absolute value of  a , denoted by | a | , is defined to be: where − a is the additive inverse of 

4356-462: A set in Euclidean space is compact if and only if it is closed and bounded.) Briefly, a closed set contains all of its boundary points , while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number. In R {\displaystyle \mathbb {R} } , sets that are closed and bounded, and therefore compact, include

4488-411: A set, it denotes its cardinality ; when applied to a matrix , it denotes its determinant . Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra , for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either

4620-568: A simple example, consider f : ( 0 , 1 ) → R {\displaystyle f:(0,1)\to \mathbb {R} } defined by f ( x ) = 1 / x {\displaystyle f(x)=1/x} . By choosing points close to 0, we can always make | f ( x ) − f ( y ) | > ε {\displaystyle |f(x)-f(y)|>\varepsilon } for any single choice of δ > 0 {\displaystyle \delta >0} , for

4752-517: Is | z | = r . {\displaystyle |z|=r.} Since the product of any complex number z {\displaystyle z} and its complex conjugate z ¯ = x − i y {\displaystyle {\bar {z}}=x-iy} , with the same absolute value, is always the non-negative real number ( x 2 + y 2 ) {\displaystyle \left(x^{2}+y^{2}\right)} ,

4884-522: Is continuous at p ∈ I {\displaystyle p\in I} if lim x → p f ( x ) = f ( p ) {\textstyle \lim _{x\to p}f(x)=f(p)} . We say that f {\displaystyle f} is a continuous map if f {\displaystyle f} is continuous at every p ∈ I {\displaystyle p\in I} . In contrast to

5016-706: Is continuous at p ∈ X {\displaystyle p\in X} if, for any ε > 0 {\displaystyle \varepsilon >0} , there exists δ > 0 {\displaystyle \delta >0} such that for all x ∈ X {\displaystyle x\in X} , | x − p | < δ {\displaystyle |x-p|<\delta } implies that | f ( x ) − f ( p ) | < ε {\displaystyle |f(x)-f(p)|<\varepsilon } . We say that f {\displaystyle f}

5148-483: Is differentiable at a {\displaystyle a} if the limit Complex analysis Complex analysis , traditionally known as the theory of functions of a complex variable , is the branch of mathematical analysis that investigates functions of complex numbers . It is helpful in many branches of mathematics, including algebraic geometry , number theory , analytic combinatorics , and applied mathematics , as well as in physics , including

5280-424: Is analytic (see next section), and two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain (if the domains are connected ). The latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with

5412-428: Is a continuous map if f {\displaystyle f} is continuous at every p ∈ X {\displaystyle p\in X} . A consequence of this definition is that f {\displaystyle f} is trivially continuous at any isolated point p ∈ X {\displaystyle p\in X} . This somewhat unintuitive treatment of isolated points

Real analysis - Misplaced Pages Continue

5544-422: Is a function from complex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain . Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane . For any complex function, the values z {\displaystyle z} from

5676-449: Is a function whose domain is a countable , totally ordered set. The domain is usually taken to be the natural numbers , although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices. Of interest in real analysis, a real-valued sequence , here indexed by the natural numbers, is a map a : N → R : n ↦

5808-585: Is a limit point of E {\displaystyle E} . A more general definition applying to f : X → R {\displaystyle f:X\to \mathbb {R} } with a general domain X ⊂ R {\displaystyle X\subset \mathbb {R} } is the following: Definition. If X {\displaystyle X} is an arbitrary subset of R {\displaystyle \mathbb {R} } , we say that f : X → R {\displaystyle f:X\to \mathbb {R} }

5940-476: Is a neighborhood of p {\displaystyle p} in X {\displaystyle X} for every neighborhood V {\displaystyle V} of f ( p ) {\displaystyle f(p)} in Y {\displaystyle Y} . We say that f {\displaystyle f} is a continuous map if f − 1 ( U ) {\displaystyle f^{-1}(U)}

6072-426: Is a positive number , and | x | = − x {\displaystyle |x|=-x} if x {\displaystyle x} is negative (in which case negating x {\displaystyle x} makes − x {\displaystyle -x} positive), and | 0 | = 0 {\displaystyle |0|=0} . For example,

6204-549: Is a positive number δ {\displaystyle \delta } such that whenever a finite sequence of pairwise disjoint sub-intervals ( x 1 , y 1 ) , ( x 2 , y 2 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),\ldots ,(x_{n},y_{n})} of I {\displaystyle I} satisfies then Absolutely continuous functions are continuous: consider

6336-413: Is a special case of multiplicativity that is often useful by itself. The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0 . It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0, +∞) . Since a real number and its opposite have the same absolute value, it is an even function , and

6468-424: Is a strictly increasing sequence of natural numbers. Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value. (This value can include the symbols ± ∞ {\displaystyle \pm \infty } when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of

6600-715: Is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely. Uniform convergence requires members of the family of functions, f n {\displaystyle f_{n}} , to fall within some error ε > 0 {\displaystyle \varepsilon >0} of f {\displaystyle f} for every value of x ∈ E {\displaystyle x\in E} , whenever n ≥ N {\displaystyle n\geq N} , for some integer N {\displaystyle N} . For

6732-828: Is a subset of the real numbers , we say a function f : X → R {\displaystyle f:X\to \mathbb {R} } is uniformly continuous on X {\displaystyle X} if, for any ε > 0 {\displaystyle \varepsilon >0} , there exists a δ > 0 {\displaystyle \delta >0} such that for all x , y ∈ X {\displaystyle x,y\in X} , | x − y | < δ {\displaystyle |x-y|<\delta } implies that | f ( x ) − f ( y ) | < ε {\displaystyle |f(x)-f(y)|<\varepsilon } . Explicitly, when

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6864-480: Is a superset of X {\displaystyle X} . This open cover is said to have a finite subcover if a finite subcollection of the U α {\displaystyle U_{\alpha }} could be found that also covers X {\displaystyle X} . Definition. A set X {\displaystyle X} in a topological space is compact if every open cover of X {\displaystyle X} has

6996-506: Is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of  | x | at  x = 0 is the interval  [−1, 1] . The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations . The second derivative of  | x | with respect to  x

7128-419: Is another example of a compact set. On the other hand, the set { 1 / n : n ∈ N } {\displaystyle \{1/n:n\in \mathbb {N} \}} is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set. The set [ 0 , ∞ ) {\displaystyle [0,\infty )} is also not compact because it

7260-524: Is called conformal (or angle-preserving) at a point u 0 ∈ U {\displaystyle u_{0}\in U} if it preserves angles between directed curves through u 0 {\displaystyle u_{0}} , as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature . The conformal property may be described in terms of

7392-428: Is closed but not bounded. For subsets of the real numbers, there are several equivalent definitions of compactness. Definition. A set E ⊂ R {\displaystyle E\subset \mathbb {R} } is compact if it is closed and bounded. This definition also holds for Euclidean space of any finite dimension, R n {\displaystyle \mathbb {R} ^{n}} , but it

7524-635: Is closely related to the notions of magnitude , distance , and norm in various mathematical and physical contexts. In 1806, Jean-Robert Argand introduced the term module , meaning unit of measure in French, specifically for the complex absolute value, and it was borrowed into English in 1866 as the Latin equivalent modulus . The term absolute value has been used in this sense from at least 1806 in French and 1857 in English. The notation | x | , with

7656-437: Is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, ( R , | ⋅ | ) {\displaystyle (\mathbb {R} ,|\cdot |)} , is a complete metric space . In a general metric space, however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that

7788-750: Is defined by | z | = Re ⁡ ( z ) 2 + Im ⁡ ( z ) 2 = x 2 + y 2 , {\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},} the Pythagorean addition of x {\displaystyle x} and y {\displaystyle y} , where Re ⁡ ( z ) = x {\displaystyle \operatorname {Re} (z)=x} and Im ⁡ ( z ) = y {\displaystyle \operatorname {Im} (z)=y} denote

7920-445: Is given below for completeness. Definition. If X {\displaystyle X} and Y {\displaystyle Y} are topological spaces, we say that f : X → Y {\displaystyle f:X\to Y} is continuous at p ∈ X {\displaystyle p\in X} if f − 1 ( V ) {\displaystyle f^{-1}(V)}

8052-461: Is hence not invertible . The real absolute value function is a piecewise linear , convex function . For both real and complex numbers the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself). The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show

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8184-530: Is holomorphic on a region Ω {\displaystyle \Omega } , then for all z 0 ∈ Ω {\displaystyle z_{0}\in \Omega } , In terms of the real and imaginary parts of the function, u and v , this is equivalent to the pair of equations u x = v y {\displaystyle u_{x}=v_{y}} and u y = − v x {\displaystyle u_{y}=-v_{x}} , where

8316-453: Is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence. For example, a sequence of continuous functions (see below ) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence

8448-506: Is in quantum mechanics as wave functions . Absolute value In mathematics , the absolute value or modulus of a real number x {\displaystyle x} , denoted | x | {\displaystyle |x|} , is the non-negative value of x {\displaystyle x} without regard to its sign . Namely, | x | = x {\displaystyle |x|=x} if x {\displaystyle x}

8580-535: Is in the domain of f {\displaystyle f} ; and (ii) f ( x ) → f ( p ) {\displaystyle f(x)\to f(p)} as x → p {\displaystyle x\to p} . The definition above actually applies to any domain E {\displaystyle E} that does not contain an isolated point , or equivalently, E {\displaystyle E} where every p ∈ E {\displaystyle p\in E}

8712-423: Is meaningless. On a compact set, it is easily shown that all continuous functions are uniformly continuous. If E {\displaystyle E} is a bounded noncompact subset of R {\displaystyle \mathbb {R} } , then there exists f : E → R {\displaystyle f:E\to \mathbb {R} } that is continuous but not uniformly continuous. As

8844-401: Is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between topological spaces (which includes metric spaces and R {\displaystyle \mathbb {R} } in particular as special cases). This definition, which extends beyond the scope of our discussion of real analysis,

8976-528: Is not valid for metric spaces in general. The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the Heine-Borel theorem . A more general definition that applies to all metric spaces uses the notion of a subsequence (see above). Definition. A set E {\displaystyle E} in a metric space is compact if every sequence in E {\displaystyle E} has

9108-574: Is nowhere real analytic . Most elementary functions, including the exponential function , the trigonometric functions , and all polynomial functions , extended appropriately to complex arguments as functions C → C {\displaystyle \mathbb {C} \to \mathbb {C} } , are holomorphic over the entire complex plane, making them entire functions , while rational functions p / q {\displaystyle p/q} , where p and q are polynomials, are holomorphic on domains that exclude points where q

9240-445: Is only pointwise. Karl Weierstrass is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications. Compactness is a concept from general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being closed and bounded . (In the context of real analysis, these notions are equivalent:

9372-483: Is open in X {\displaystyle X} for every U {\displaystyle U} open in Y {\displaystyle Y} . (Here, f − 1 ( S ) {\displaystyle f^{-1}(S)} refers to the preimage of S ⊂ Y {\displaystyle S\subset Y} under f {\displaystyle f} .) Definition. If X {\displaystyle X}

9504-459: Is particularly concerned with analytic functions of a complex variable, that is, holomorphic functions . The concept can be extended to functions of several complex variables . Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler , Gauss , Riemann , Cauchy , Gösta Mittag-Leffler , Weierstrass , and many more in

9636-421: Is thus always either a positive number or zero , but never negative . When x {\displaystyle x} itself is negative ( x < 0 {\displaystyle x<0} ), then its absolute value is necessarily positive ( | x | = − x > 0 {\displaystyle |x|=-x>0} ). From an analytic geometry point of view,

9768-495: Is zero everywhere except zero, where it does not exist. As a generalised function , the second derivative may be taken as two times the Dirac delta function . The antiderivative (indefinite integral ) of the real absolute value function is where C is an arbitrary constant of integration . This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable ( holomorphic ) functions, which

9900-502: Is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions . On the other hand, the functions z ↦ ℜ ( z ) {\displaystyle z\mapsto \Re (z)} , z ↦ | z | {\displaystyle z\mapsto |z|} , and z ↦ z ¯ {\displaystyle z\mapsto {\bar {z}}} are not holomorphic anywhere on

10032-483: The Euclidean norm or sup norm of a vector in R n {\displaystyle \mathbb {R} ^{n}} , although double vertical bars with subscripts ( ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} and ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }} , respectively) are

10164-504: The Jacobian derivative matrix of a coordinate transformation . The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix ( orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions,

10296-418: The Pythagorean theorem : for any complex number z = x + i y , {\displaystyle z=x+iy,} where x {\displaystyle x} and y {\displaystyle y} are real numbers, the absolute value or modulus of z {\displaystyle z} is denoted | z | {\displaystyle |z|} and

10428-557: The field of complex numbers is algebraically closed . If a function is holomorphic throughout a connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions, such as the Riemann zeta function , which are initially defined in terms of infinite sums that converge only on limited domains to almost

10560-557: The intermediate value theorem and the mean value theorem . However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in functional analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces and positive operators . Also, mathematicians consider real and imaginary parts of complex sequences, or by pointwise evaluation of operator sequences. Many of

10692-415: The metric or distance function d : R × R → R ≥ 0 {\displaystyle d:\mathbb {R} \times \mathbb {R} \to \mathbb {R} _{\geq 0}} using the absolute value function as d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} , the real numbers become

10824-766: The n th derivative need not imply the existence of the ( n + 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of analyticity , meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on Ω {\displaystyle \Omega } can be approximated arbitrarily well by polynomials in some neighborhood of every point in Ω {\displaystyle \Omega } . This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are nowhere analytic; see Non-analytic smooth function § A smooth function which

10956-442: The square root symbol represents the unique positive square root , when applied to a positive number, it follows that | x | = x 2 . {\displaystyle |x|={\sqrt {x^{2}}}.} This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers. The absolute value has the following four fundamental properties (

11088-404: The (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types. One of the central tools in complex analysis is the line integral . The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by

11220-493: The 20th century. Complex analysis, in particular the theory of conformal mappings , has many physical applications and is also used throughout analytic number theory . In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions . Another important application of complex analysis is in string theory which examines conformal invariants in quantum field theory . A complex function

11352-417: The absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers , the quaternions , ordered rings , fields and vector spaces . The absolute value

11484-520: The absolute value of a complex number z {\displaystyle z} is the square root of z ⋅ z ¯ , {\displaystyle z\cdot {\overline {z}},} which is therefore called the absolute square or squared modulus of z {\displaystyle z} : | z | = z ⋅ z ¯ . {\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.} This generalizes

11616-410: The absolute value of a real number is that number's distance from zero along the real number line , and more generally the absolute value of the difference of two real numbers (their absolute difference ) is the distance between them. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since

11748-418: The absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points and in Euclidean n -space is defined as: This can be seen as a generalisation, since for a 1 {\displaystyle a_{1}} and b 1 {\displaystyle b_{1}} real, i.e. in a 1-space, according to

11880-403: The alternative definition for reals: | x | = x ⋅ x {\textstyle |x|={\sqrt {x\cdot x}}} . The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity | z | 2 = | z 2 | {\displaystyle |z|^{2}=|z^{2}|}

12012-415: The alternative definition of the absolute value, and for a = a 1 + i a 2 {\displaystyle a=a_{1}+ia_{2}} and b = b 1 + i b 2 {\displaystyle b=b_{1}+ib_{2}} complex numbers, i.e. in a 2-space, The above shows that the "absolute value"-distance, for real and complex numbers, agrees with

12144-466: The analytic properties such as power series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality ) do not carry over. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions. A major application of certain complex spaces

12276-412: The branches of hydrodynamics , thermodynamics , quantum mechanics , and twistor theory . By extension, use of complex analysis also has applications in engineering fields such as nuclear , aerospace , mechanical and electrical engineering . As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic ), complex analysis

12408-468: The case n = 1 in this definition. The collection of all absolutely continuous functions on I is denoted AC( I ). Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral. The notion of the derivative of a function or differentiability originates from

12540-844: The case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence , that need to be distinguished. Roughly speaking, pointwise convergence of functions f n {\displaystyle f_{n}} to a limiting function f : E → R {\displaystyle f:E\to \mathbb {R} } , denoted f n → f {\displaystyle f_{n}\rightarrow f} , simply means that given any x ∈ E {\displaystyle x\in E} , f n ( x ) → f ( x ) {\displaystyle f_{n}(x)\to f(x)} as n → ∞ {\displaystyle n\to \infty } . In contrast, uniform convergence

12672-426: The choice of δ {\displaystyle \delta } may depend on both ε {\displaystyle \varepsilon } and p {\displaystyle p} . In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p {\displaystyle p}

12804-516: The closed path is always zero, as is stated by the Cauchy integral theorem . The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in Cauchy's integral formula ). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is applicable (see methods of contour integration ). A "pole" (or isolated singularity ) of

12936-402: The complex absolute value function is not. The following two formulae are special cases of the chain rule : d d x f ( | x | ) = x | x | ( f ′ ( | x | ) ) {\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))} if the absolute value is inside

13068-863: The complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below). An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy–Riemann conditions . If f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } , defined by f ( z ) = f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)} , where x , y , u ( x , y ) , v ( x , y ) ∈ R {\displaystyle x,y,u(x,y),v(x,y)\in \mathbb {R} } ,

13200-428: The complex-valued equivalent to Taylor series , but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials. A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouville's theorem . It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that

13332-491: The concept of approximating a function near a given point using the "best" linear approximation. This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a {\displaystyle a} , and the slope of the line is the derivative of the function at a {\displaystyle a} . A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} }

13464-457: The context of complex analysis, the derivative of f {\displaystyle f} at z 0 {\displaystyle z_{0}} is defined to be Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist,

13596-406: The corresponding definition of the limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} decreases without bound , lim x → − ∞ f ( x ) {\textstyle \lim _{x\to -\infty }f(x)} . Sometimes, it is useful to conclude that a sequence converges, even though

13728-399: The definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin . This can be computed using

13860-483: The definition of the limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} increases without bound , notated lim x → ∞ f ( x ) {\textstyle \lim _{x\to \infty }f(x)} . Reversing the inequality x ≥ M {\displaystyle x\geq M} to x ≤ M {\displaystyle x\leq M} gives

13992-409: The definition, is to ensure that lim x → x 0 f ( x ) = L {\textstyle \lim _{x\to x_{0}}f(x)=L} does not imply anything about the value of f ( x 0 ) {\displaystyle f(x_{0})} itself. Actually, x 0 {\displaystyle x_{0}} does not even need to be in

14124-988: The domain and their images f ( z ) {\displaystyle f(z)} in the range may be separated into real and imaginary parts: where x , y , u ( x , y ) , v ( x , y ) {\displaystyle x,y,u(x,y),v(x,y)} are all real-valued. In other words, a complex function f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } may be decomposed into i.e., into two real-valued functions ( u {\displaystyle u} , v {\displaystyle v} ) of two real variables ( x {\displaystyle x} , y {\displaystyle y} ). Similarly, any complex-valued function f on an arbitrary set X (is isomorphic to, and therefore, in that sense, it) can be considered as an ordered pair of two real-valued functions : (Re f , Im f ) or, alternatively, as

14256-495: The domain of f {\displaystyle f} in order for lim x → x 0 f ( x ) {\textstyle \lim _{x\to x_{0}}f(x)} to exist. In a slightly different but related context, the concept of a limit applies to the behavior of a sequence ( a n ) {\displaystyle (a_{n})} when n {\displaystyle n} becomes large. Definition. Let (

14388-507: The domain of f {\displaystyle f} ) is a real number that is less than δ {\displaystyle \delta } away from x 0 {\displaystyle x_{0}} but distinct from x 0 {\displaystyle x_{0}} . The purpose of the last stipulation, which corresponds to the condition 0 < | x − x 0 | {\displaystyle 0<|x-x_{0}|} in

14520-518: The empty set, any finite number of points, closed intervals , and their finite unions. However, this list is not exhaustive; for instance, the set { 1 / n : n ∈ N } ∪ { 0 } {\displaystyle \{1/n:n\in \mathbb {N} \}\cup \{0}\} is a compact set; the Cantor ternary set C ⊂ [ 0 , 1 ] {\displaystyle {\mathcal {C}}\subset [0,1]}

14652-718: The end of the 17th century, for building infinitesimal calculus . For sequences, the concept was introduced by Cauchy , and made rigorous, at the end of the 19th century by Bolzano and Weierstrass , who gave the modern ε-δ definition , which follows. Definition. Let f {\displaystyle f} be a real-valued function defined on E ⊂ R {\displaystyle E\subset \mathbb {R} } . We say that f ( x ) {\displaystyle f(x)} tends to L {\displaystyle L} as x {\displaystyle x} approaches x 0 {\displaystyle x_{0}} , or that

14784-470: The entire complex plane. Sometimes, as in the case of the natural logarithm , it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface . All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension in which

14916-407: The first case and where f ( x ) = 0 {\textstyle f(x)=0} in the second case. The absolute value is closely related to the idea of distance . As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally,

15048-702: The following way: We say that f ( x ) → L {\displaystyle f(x)\to L} as x → x 0 {\displaystyle x\to x_{0}} , when, given any positive number ε {\displaystyle \varepsilon } , no matter how small, we can always find a δ {\displaystyle \delta } , such that we can guarantee that f ( x ) {\displaystyle f(x)} and L {\displaystyle L} are less than ε {\displaystyle \varepsilon } apart, as long as x {\displaystyle x} (in

15180-405: The four fundamental properties above. Two other useful properties concerning inequalities are: These relations may be used to solve inequalities involving absolute values. For example: The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers. Since the complex numbers are not ordered ,

15312-1278: The limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} approaches x 0 {\displaystyle x_{0}} is L {\displaystyle L} if, for any ε > 0 {\displaystyle \varepsilon >0} , there exists δ > 0 {\displaystyle \delta >0} such that for all x ∈ E {\displaystyle x\in E} , 0 < | x − x 0 | < δ {\displaystyle 0<|x-x_{0}|<\delta } implies that | f ( x ) − L | < ε {\displaystyle |f(x)-L|<\varepsilon } . We write this symbolically as f ( x ) → L     as     x → x 0 , {\displaystyle f(x)\to L\ \ {\text{as}}\ \ x\to x_{0},} or as lim x → x 0 f ( x ) = L . {\displaystyle \lim _{x\to x_{0}}f(x)=L.} Intuitively, this definition can be thought of in

15444-421: The notion of open covers and subcovers , which is applicable to topological spaces (and thus to metric spaces and R {\displaystyle \mathbb {R} } as special cases). In brief, a collection of open sets U α {\displaystyle U_{\alpha }} is said to be an open cover of set X {\displaystyle X} if the union of these sets

15576-411: The ordering of the real numbers is total , and the real numbers have the least upper bound property : Every nonempty subset of R {\displaystyle \mathbb {R} } that has an upper bound has a least upper bound that is also a real number. These order-theoretic properties lead to a number of fundamental results in real analysis, such as the monotone convergence theorem ,

15708-615: The prototypical example of a metric space . The topology induced by metric d {\displaystyle d} turns out to be identical to the standard topology induced by order < {\displaystyle <} . Theorems like the intermediate value theorem that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in R {\displaystyle \mathbb {R} } only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods. A sequence

15840-689: The range of an entire function f {\displaystyle f} , then f {\displaystyle f} is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset. In mathematics , a conformal map is a function that locally preserves angles , but not necessarily lengths. More formally, let U {\displaystyle U} and V {\displaystyle V} be open subsets of R n {\displaystyle \mathbb {R} ^{n}} . A function f : U → V {\displaystyle f:U\to V}

15972-510: The rational numbers Q {\displaystyle \mathbb {Q} } ) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the least upper bound property (see below). The real numbers have various lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an ordered field , in which sums and products of positive numbers are also positive. Moreover,

16104-535: The real and imaginary parts of z {\displaystyle z} , respectively. When the imaginary part y {\displaystyle y} is zero, this coincides with the definition of the absolute value of the real number x {\displaystyle x} . When a complex number z {\displaystyle z} is expressed in its polar form as z = r e i θ , {\displaystyle z=re^{i\theta },} its absolute value

16236-401: The relationship between these two functions: or and for x ≠ 0 , Let s , t ∈ R {\displaystyle s,t\in \mathbb {R} } , then and The real absolute value function has a derivative for every x ≠ 0 , but is not differentiable at x = 0 . Its derivative for x ≠ 0 is given by the step function : The real absolute value function

16368-761: The requirements for f {\displaystyle f} to have a limit at a point p {\displaystyle p} , which do not constrain the behavior of f {\displaystyle f} at p {\displaystyle p} itself, the following two conditions, in addition to the existence of lim x → p f ( x ) {\textstyle \lim _{x\to p}f(x)} , must also hold in order for f {\displaystyle f} to be continuous at p {\displaystyle p} : (i) f {\displaystyle f} must be defined at p {\displaystyle p} , i.e., p {\displaystyle p}

16500-660: The result positive. Now, since − 1 ⋅ x ≤ | x | {\displaystyle -1\cdot x\leq |x|} and + 1 ⋅ x ≤ | x | {\displaystyle +1\cdot x\leq |x|} , it follows that, whichever of ± 1 {\displaystyle \pm 1} is the value of s {\displaystyle s} , one has s ⋅ x ≤ | x | {\displaystyle s\cdot x\leq |x|} for all real x {\displaystyle x} . Consequently, |

16632-586: The sequence is bounded if and only if it is convergent. In addition to sequences of numbers, one may also speak of sequences of functions on E ⊂ R {\displaystyle E\subset \mathbb {R} } , that is, infinite, ordered families of functions f n : E → R {\displaystyle f_{n}:E\to \mathbb {R} } , denoted ( f n ) n = 1 ∞ {\displaystyle (f_{n})_{n=1}^{\infty }} , and their convergence properties. However, in

16764-442: The standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows: A real valued function d on

16896-911: The subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see Looman–Menchoff theorem ). Holomorphic functions exhibit some remarkable features. For instance, Picard's theorem asserts that the range of an entire function can take only three possible forms: C {\displaystyle \mathbb {C} } , C ∖ { z 0 } {\displaystyle \mathbb {C} \setminus \{z_{0}\}} , or { z 0 } {\displaystyle \{z_{0}\}} for some z 0 ∈ C {\displaystyle z_{0}\in \mathbb {C} } . In other words, if two distinct complex numbers z {\displaystyle z} and w {\displaystyle w} are not in

17028-408: The theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties. As a topological space , the real numbers has a standard topology , which is the order topology induced by order < {\displaystyle <} . Alternatively, by defining

17160-401: The value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z 0 {\displaystyle z_{0}} in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are infinitely differentiable , whereas the existence of

17292-463: The value to which it converges is unknown or irrelevant. In these cases, the concept of a Cauchy sequence is useful. Definition. Let ( a n ) {\displaystyle (a_{n})} be a real-valued sequence. We say that ( a n ) {\displaystyle (a_{n})} is a Cauchy sequence if, for any ε > 0 {\displaystyle \varepsilon >0} , there exists

17424-515: The whole set of real numbers, an open interval I = ( a , b ) = { x ∈ R ∣ a < x < b } , {\displaystyle I=(a,b)=\{x\in \mathbb {R} \mid a<x<b\},} or a closed interval I = [ a , b ] = { x ∈ R ∣ a ≤ x ≤ b } . {\displaystyle I=[a,b]=\{x\in \mathbb {R} \mid a\leq x\leq b\}.} Here,

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