Complex Numbers - Revision history

Lila at 16:06, 3 November 2021

2021-11-03T16:06:03Z

Lila at 16:04, 3 November 2021

2021-11-03T16:04:58Z

Lila at 16:04, 3 November 2021

2021-11-03T16:04:07Z

Lila: /* Integer and fractional exponents */

2021-11-03T16:02:55Z

Integer and fractional exponents

Lila: /* Integer and fractional exponents */

2021-11-03T16:02:05Z

Integer and fractional exponents

Lila: /* Exponentiation */

2021-11-03T16:01:00Z

Exponentiation

Lila: /* Complex logarithm */

2021-11-03T16:00:23Z

Complex logarithm

Lila: /* Exponential function */

2021-11-03T15:58:44Z

Exponential function

Lila: /* Square root */

2021-11-03T15:57:16Z

Square root

Lila: /* Reciprocal and division */

2021-11-03T15:56:39Z

Reciprocal and division

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 [[File:Illustration of a complex number.svg|right|thumb|upright=1.05|An illustration of the complex number {{math|1=''z'' = ''x'' + ''iy''}} on the complex plane. The real part is {{mvar|x}}, and its imaginary part is {{mvar|y}}.]]
-A complex number is a number of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are real numbers, and {{math|''i''}} is an indeterminate satisfying {{math|1=''i''<sup>2</sup> = −1}}. For example, {{math|2 + 3''i''}} is a complex number.<ref>{{cite book|title=College algebra |url=https://archive.org/details/collegealgebrawi00axle |url-access=limited |last=Axler |first=Sheldon |page=[https://archive.org/details/collegealgebrawi00axle/page/n285 262]|publisher=Wiley|year=2010|isbn=9780470470770 }}</ref><ref name=":1" />
+A complex number is a number of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are real numbers, and {{math|''i''}} is an indeterminate satisfying {{math|1=''i''<sup>2</sup> = −1}}. For example, {{math|2 + 3''i''}} is a complex number.
 This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate {{math|''i''}}, for which the relation {{math|''i''<sup>2</sup> + 1 = 0}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{math|''i''<sup>2</sup> + 1 = 0}} induces the equalities {{math|''i''<sup>4''k''</sup> = 1, ''i''<sup>4''k''+1</sup> = ''i'', ''i''<sup>4''k''+2</sup> = −1,}} and {{math|''i''<sup>4''k''+3</sup> = −''i'',}} which hold for all integers {{mvar|k}}; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in {{mvar|i}}, again of the form {{math|1=''a'' + ''bi''}} with real coefficients {{mvar|a, b.}}
@@ Line 29: / Line 29: @@
 A real number {{mvar|a}} can be regarded as a complex number {{math|''a'' + 0''i''}}, whose imaginary part is 0. A purely imaginary number {{math|''bi''}} is a complex number {{math|0 + ''bi''}}, whose real part is zero. As with polynomials, it is common to write {{mvar|a}} for {{math|''a'' + 0''i''}} and {{math|''bi''}} for {{math|0 + ''bi''}}. Moreover, when the imaginary part is negative, that is, {{math|1=''b'' = −''{{!}}b{{!}}'' < 0}}, it is common to write {{math|''a'' − ''{{!}}b{{!}}i''}} instead of {{math|''a'' + (−''{{!}}b{{!}}'')''i''}}; for example, for {{math|1=''b'' = −4}}, {{math|3 − 4''i''}} can be written instead of {{math|3 + (−4)''i''}}.
-Since the multiplication of the indeterminate {{math|''i''}} and a real is commutative in polynomials with real coefficients, the polynomial {{math|''a'' + ''bi''}} may be written as {{math|''a'' + ''ib''.}} This is often expedient for imaginary parts denoted by expressions, for example, when {{mvar|b}} is a radical.<ref>See {{harv|Ahlfors|1979}}.</ref>
+Since the multiplication of the indeterminate {{math|''i''}} and a real is commutative in polynomials with real coefficients, the polynomial {{math|''a'' + ''bi''}} may be written as {{math|''a'' + ''ib''.}} This is often expedient for imaginary parts denoted by expressions, for example, when {{mvar|b}} is a radical.
 The real part of a complex number {{mvar|z}} is denoted by {{math|Re(''z'')}}, <math>\mathcal{Re}(z)</math>, or <math>\mathfrak{R}(z)</math>; the imaginary part of a complex number {{mvar|z}} is denoted by {{math|Im(''z'')}}, <math>\mathcal{Im}(z)</math>, or <math>\mathfrak{I}(z).</math> For example,
@@ Line 65: / Line 65: @@
 By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane.
-The ''argument'' of {{mvar|z}} (in many applications referred to as the "phase" {{mvar|φ}})<ref name=":2" /> is the angle of the radius {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}. As with the modulus, the argument can be found from the rectangular form {{mvar|x + yi}} by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the {{math|arg}}-function, (−''π'', ''π''], and avoids a more subtle case-by-case analysis
+The ''argument'' of {{mvar|z}} (in many applications referred to as the "phase" {{mvar|φ}}) is the angle of the radius {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}. As with the modulus, the argument can be found from the rectangular form {{mvar|x + yi}} by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the {{math|arg}}-function, (−''π'', ''π''], and avoids a more subtle case-by-case analysis
 <math display=block>\varphi = \arg (x+yi) = \begin{cases}

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 ===Conjugate===
-[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate {{mvar|{{overline|z}}}} in the complex plane]]
+[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate <math>\overline{z}</math> in the complex plane]]
 The ''complex conjugate'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is given by {{math|''x'' − ''yi''}}. It is denoted by either <math>\overline{z}</math> or {{math|''z''*}}. This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

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 has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions {{math|−1 + 3''i''}} and {{math|−1 − 3''i''}}.
-Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule {{math|1=''i''<sup>2</sup> = −1}} combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {{math|{{mset|1, ''i''}}}} as a standard basis.
+Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule {{math|1=''i''<sup>2</sup> = −1}} combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {{math|{1, ''i''}} as a standard basis.
 This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.

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 ====Integer and fractional exponents====
-If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then {{math|''z''{{sup|''n''}}}} is well defined, and the exponentiation formula simplifies to de Moivre's formula:
+If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then <math>z^n</math> is well defined, and the exponentiation formula simplifies to de Moivre's formula:
 <math display=block> z^{n}=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>

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 ====Integer and fractional exponents====
-{{Visualisation complex number roots|1=upright=1.35}}
-If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then {{math|''z''{{sup|''n''}}}} is well defined, and the exponentiation formula simplifies to [[de Moivre's formula]]:
+If, in the preceding formula, {{mvar|t}} is an integer, then the sine and the cosine are independent of {{mvar|k}}. Thus, if the exponent {{mvar|n}} is an integer, then {{math|''z''{{sup|''n''}}}} is well defined, and the exponentiation formula simplifies to de Moivre's formula:
 <math display=block> z^{n}=(r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>
-The {{mvar|n}} [[nth root|{{mvar|n}}th roots]] of a complex number {{mvar|z}} are given by
+The {{mvar|n}} {{mvar|n}}th roots of a complex number {{mvar|z}} are given by
 <math display=block>z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)</math>
 for {{math|0 ≤ ''k'' ≤ ''n'' − 1}}. (Here <math>\sqrt[n]r</math> is the usual (positive) {{mvar|n}}th root of the positive real number {{mvar|r}}.) Because sine and cosine are periodic, other integer values of {{mvar|k}} do not give other values.
-While the {{mvar|n}}th root of a positive real number {{mvar|r}} is chosen to be the ''positive'' real number {{mvar|c}} satisfying {{math|1=''c''<sup>''n''</sup> = ''r''}}, there is no natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. Therefore, the {{mvar|n}}th root is a [[multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}. This implies that, contrary to the case of positive real numbers, one has
+While the {{mvar|n}}th root of a positive real number {{mvar|r}} is chosen to be the ''positive'' real number {{mvar|c}} satisfying {{math|1=''c''<sup>''n''</sup> = ''r''}}, there is no natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. Therefore, the {{mvar|n}}th root is a {{mvar|n}}-valued function of {{mvar|z}}. This implies that, contrary to the case of positive real numbers, one has
 <math display=block>(z^n)^{1/n} \ne z,</math>
 since the left-hand side consists of {{mvar|n}} values, and the right-hand side is a single value.

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 where {{math|ln}} denotes the natural logarithm.
-It seems natural to extend this formula to complex values of {{mvar|x}}, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a [[multivalued function]].
+It seems natural to extend this formula to complex values of {{mvar|x}}, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.
 It follows that if {{mvar|z}} is as above, and if {{mvar|t}} is another complex number, then the ''exponentiation'' is the multivalued function

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 ===Complex logarithm===
-In the real case, the [[natural logarithm]] can be defined as the [[inverse function|inverse]]
+In the real case, the natural logarithm can be defined as the inverse
-<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number <math>z\in \Complex^\times</math> is written in [[polar form]]
+<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number <math>z\in \Complex^\times</math> is written in polar form
 <math display=block> z = r(\cos \varphi + i\sin \varphi )</math>
 with <math>r, \varphi \in \R ,</math> then with
 <math display=block> \ln z = \ln r + i \varphi </math>
-as [[complex logarithm]] one has a proper inverse:
+as complex logarithm one has a proper inverse:
 <math display=block> \exp \ln z = \exp(\ln r + i \varphi ) = r \exp i \varphi = r(\cos \varphi + i\sin \varphi ) = z .</math>
 However, because cosine and sine are periodic functions, the addition of an integer multiple of {{math|2''π''}} to {{mvar|φ}} does not change {{mvar|z}}. For example, {{math|1=''e''<sup>''iπ''</sup> = ''e''<sup>3''iπ''</sup> = −1}} , so both {{mvar|iπ}} and {{math|3''iπ''}} are possible values for the natural logarithm of {{math|−1}}.
-Therefore, if the complex logarithm is not to be defined as a [[multivalued function]]
+Therefore, if the complex logarithm is not to be defined as a multivalued function
 <math display=block> \ln z = \left\{ \ln r + i (\varphi + 2\pi k) \mid k \in \Z \right\},</math>
-one has to use a [[branch cut]] and to restrict the [[codomain]], resulting in the [[bijective]] function
+one has to use a branch cut and to restrict the codomain, resulting in the bijective function
 <math display=block>\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .</math>
-If <math>z \in \Complex \setminus \left( -\R_{\ge 0} \right)</math> is not a non-positive real number (a positive or a non-real number), the resulting [[principal value]] of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an [[analytic function]] outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math>z \in -\R^+ </math>, where the principal value is {{math|1=ln ''z'' = ln(−''z'') + ''iπ''}}.{{efn|However for another inverse function of the complex exponential function (and not the above defined principal value), the branch cut could be taken at any other [[Line (geometry)#Ray|ray]] thru the origin.}}
+If <math>z \in \Complex \setminus \left( -\R_{\ge 0} \right)</math> is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math>z \in -\R^+ </math>, where the principal value is {{math|1=ln ''z'' = ln(−''z'') + ''iπ''}}.
 ===Exponentiation===

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 ===Exponential function===
-The [[exponential function]] <math>\exp \colon \Complex \to \Complex ; z \mapsto \exp z </math> can be defined for every complex number {{mvar|z}} by the [[power series]]
+The exponential function <math>\exp \colon \Complex \to \Complex ; z \mapsto \exp z </math> can be defined for every complex number {{mvar|z}} by the power series
 <math display=block>\exp z= \sum_{n=0}^\infty \frac {z^n}{n!},</math>
-which has an infinite [[radius of convergence]].
+which has an infinite radius of convergence.
-The value at {{math|1}} of the exponential function is [[Euler's number]]
+The value at {{math|1}} of the exponential function is Euler's number
 <math display=block>e = \exp 1 = \sum_{n=0}^\infty \frac1{n!}\approx 2.71828.</math>
 If {{mvar|z}} is real, one has
 <math>\exp z=e^z.</math>
-[[Analytic continuation]] allows extending this equality for every complex value of {{mvar|z}}, and thus to define the complex exponentiation with base {{mvar|e}} as
+Analytic continuation allows extending this equality for every complex value of {{mvar|z}}, and thus to define the complex exponentiation with base {{mvar|e}} as
 <math display=block>e^z=\exp z.</math>
 ====Functional equation====
-The exponential function satisfies the [[functional equation]] <math>e^{z+t}=e^ze^t.</math>
+The exponential function satisfies the functional equation <math>e^{z+t}=e^ze^t.</math>
-This can be proved either by comparing the power series expansion of both members or by applying [[analytic continuation]] from the restriction of the equation to real arguments.
+This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.
 ====Euler's formula====
-[[Euler's formula]] states that, for any real number {{mvar|y}},
+Euler's formula states that, for any real number {{mvar|y}},
 <math display=block>e^{iy} = \cos y + i\sin y .</math>

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 ===Square root===
-{{see also|Square root#Square roots of negative and complex numbers|l1=Square roots of negative and complex numbers}}
 The square roots of {{math|''a'' + ''bi''}} (with {{math|''b'' ≠ 0}}) are <math> \pm (\gamma + \delta i)</math>, where

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 since ''non-zero'' implies that {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} is greater than zero.
-This can be used to express a division of an arbitrary complex number {{math|''w'' {{=}} ''u'' + ''vi''}} by a non-zero complex number {{mvar|z}} as
+This can be used to express a division of an arbitrary complex number {{math|''w'' = ''u'' + ''vi''}} by a non-zero complex number {{mvar|z}} as
 <math display=block>\frac {w}{z}= w\cdot \frac {1}{z}= (u+vi)\cdot \left(\frac{x}{x^2+y^2} -\frac{y}{x^2+y^2}i\right)= \frac{(ux+vy)+(vx-uy)i} {x^2+y^2}.</math>