The complex plane - Revision history

Khanh at 23:10, 28 October 2021

2021-10-28T23:10:46Z

Khanh at 15:36, 10 October 2021

2021-10-10T15:36:29Z

Lila: /* The complex plane */

2021-10-10T15:28:48Z

The complex plane

Lila at 15:28, 10 October 2021

2021-10-10T15:28:19Z

Lila: /* Polar form */

2021-10-10T15:27:46Z

Polar form

Lila: /* Polar form */

2021-10-10T15:26:58Z

Polar form

Lila: /* The complex plane */

2021-10-10T15:26:24Z

The complex plane

Lila at 15:25, 10 October 2021

2021-10-10T15:25:04Z

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T22:28:37Z

fixing link to go straight to pdf

Rylee.taylor: added The complex plane notes by professor Esparza

2020-08-11T22:03:37Z

added The complex plane notes by professor Esparza

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/ The complex plane]. Written notes created by Professor Esparza, UTSA.

← Older revision		Revision as of 23:10, 28 October 2021
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	==Resources==		==Resources==
−	* [https://en.wikibooks.org/wiki/Calculus/Complex_numbers Complex Numbers], Wikibooks: Calculus
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/Esparza%201093%20Notes%20Week%2012.pdf The complex plane]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/Esparza%201093%20Notes%20Week%2012.pdf The complex plane]. Written notes created by Professor Esparza, UTSA.
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Complex_numbers Complex Numbers, Wikibooks: Calculus] under a CC BY-SA license

@@ Line 44: / Line 44: @@
 Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.
-Using [[w:List of trigonometric identities#Angle sum and difference identities|sum and difference identities]] its possible to obtain that
+Using sum and difference identities its possible to obtain that
 :<math>r_1e^{i\varphi_1}\cdot r_2e^{i\varphi_2}=r_1r_2e^{i(\varphi_1+\varphi_2)}</math>
@@ Line 52: / Line 52: @@
 :<math>\frac{r_1e^{i\varphi_1}}{r_2e^{i\varphi_2}}=\frac{r_1}{r_2}\cdot e^{i(\varphi_1-\varphi_2)}</math>
-Exponentiation with integer exponents; according to [[w:De Moivre's formula|de Moivre's formula]],
+Exponentiation with integer exponents; according to de Moivre's formula,
 :<math>\big(re^{i\varphi}\big)^n=r^ne^{ni\varphi}</math>
-Exponentiation with arbitrary complex exponents is discussed in the article on [[exponentiation]].
+Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.
 The addition of two complex numbers is just the addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.
@@ Line 79: / Line 79: @@
 :<math>|z\cdot w|=|z|\cdot|w|</math>
-for all complex numbers <math>z,w</math> . It then follows, for example, that <math>|1|=1</math> and <math>\left|\frac{z}{w}\right|=\frac{|z|}{|w|}</math> . By defining the '''distance''' function <math>d(z,w)=|z-w|</math> we turn the set of complex numbers into a [[w:metric space|metric space]] and we can therefore talk about limits and continuity.
+for all complex numbers <math>z,w</math> . It then follows, for example, that <math>|1|=1</math> and <math>\left|\frac{z}{w}\right|=\frac{|z|}{|w|}</math> . By defining the '''distance''' function <math>d(z,w)=|z-w|</math> we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.
 The '''complex conjugate''' of the complex number <math>z=a+bi</math> is defined to be <math>a-bi</math> , written as <math>\bar z</math> or <math>z^*</math> . As seen in the figure, <math>\bar z</math> is the "reflection" of <math>z</math> about the real axis. The following can be checked:

@@ Line 1: / Line 1: @@
 ==The complex plane==
 [[File:ComplexPlane.svg|ComplexPlane]]
 A complex number <math>z</math> can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the '''complex plane''' or '''Argand diagram'''. The point and hence the complex number <math>z</math> can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part <math>x=\text{Re}(z)</math> and the imaginary part <math>y=\text{Im}(z)</math> . The representation of a complex number by its Cartesian coordinates is called the ''Cartesian form'' or ''rectangular form'' or ''algebraic form'' of that complex number.

← Older revision		Revision as of 15:28, 10 October 2021
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		+	==The complex plane==
	[[File:ComplexPlane.svg\|ComplexPlane]]		[[File:ComplexPlane.svg\|ComplexPlane]]
−	~~==The complex plane==~~
	A complex number <math>z</math> can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the '''complex plane''' or '''Argand diagram'''. The point and hence the complex number <math>z</math> can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part <math>x=\text{Re}(z)</math> and the imaginary part <math>y=\text{Im}(z)</math> . The representation of a complex number by its Cartesian coordinates is called the ''Cartesian form'' or ''rectangular form'' or ''algebraic form'' of that complex number.		A complex number <math>z</math> can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the '''complex plane''' or '''Argand diagram'''. The point and hence the complex number <math>z</math> can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part <math>x=\text{Re}(z)</math> and the imaginary part <math>y=\text{Im}(z)</math> . The representation of a complex number by its Cartesian coordinates is called the ''Cartesian form'' or ''rectangular form'' or ''algebraic form'' of that complex number.

@@ Line 4: / Line 4: @@
 ===Polar form===
-[[File:Complex vector.svg|Complex_vector]]
+[[File:Complex vector.svg|thumb|Complex number in polar coordinates]]
 Alternatively, the complex number <math>z</math> can be specified by polar coordinates. The polar coordinates are <math>r=|z|\ge0</math> , called the '''absolute value''' or '''modulus''', and <math>\phi=\arg(z)</math> , called the '''argument''' of <math>z</math> . For <math>r=0</math> any value of <math>\varphi</math> describes the same number. To get a unique representation, a conventional choice is to set <math>\arg(0)=0</math> . For <math>r>0</math> the argument <math>\varphi</math> is unique modulo <math>2\pi</math> ; that is, if any two values of the complex argument differ by an exact integer multiple of <math>2\pi</math> , they are considered equivalent. To get a unique representation, a conventional choice is to limit <math>\varphi</math> to the interval <math>(-\pi,\pi]</math> i.e. <math>-\pi<\varphi\le\pi</math> . The representation of a complex number by its polar coordinates is called the ''polar form'' of the complex number.

← Older revision		Revision as of 15:26, 10 October 2021
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	===Polar form===		===Polar form===
		+	[[File:Complex vector.svg\|Complex_vector]]
	Alternatively, the complex number <math>z</math> can be specified by polar coordinates. The polar coordinates are <math>r=\|z\|\ge0</math> , called the '''absolute value''' or '''modulus''', and <math>\phi=\arg(z)</math> , called the '''argument''' of <math>z</math> . For <math>r=0</math> any value of <math>\varphi</math> describes the same number. To get a unique representation, a conventional choice is to set <math>\arg(0)=0</math> . For <math>r>0</math> the argument <math>\varphi</math> is unique modulo <math>2\pi</math> ; that is, if any two values of the complex argument differ by an exact integer multiple of <math>2\pi</math> , they are considered equivalent. To get a unique representation, a conventional choice is to limit <math>\varphi</math> to the interval <math>(-\pi,\pi]</math> i.e. <math>-\pi<\varphi\le\pi</math> . The representation of a complex number by its polar coordinates is called the ''polar form'' of the complex number.		Alternatively, the complex number <math>z</math> can be specified by polar coordinates. The polar coordinates are <math>r=\|z\|\ge0</math> , called the '''absolute value''' or '''modulus''', and <math>\phi=\arg(z)</math> , called the '''argument''' of <math>z</math> . For <math>r=0</math> any value of <math>\varphi</math> describes the same number. To get a unique representation, a conventional choice is to set <math>\arg(0)=0</math> . For <math>r>0</math> the argument <math>\varphi</math> is unique modulo <math>2\pi</math> ; that is, if any two values of the complex argument differ by an exact integer multiple of <math>2\pi</math> , they are considered equivalent. To get a unique representation, a conventional choice is to limit <math>\varphi</math> to the interval <math>(-\pi,\pi]</math> i.e. <math>-\pi<\varphi\le\pi</math> . The representation of a complex number by its polar coordinates is called the ''polar form'' of the complex number.

← Older revision		Revision as of 15:25, 10 October 2021
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		+	==The complex plane==
		+	A complex number <math>z</math> can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the '''complex plane''' or '''Argand diagram'''. The point and hence the complex number <math>z</math> can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part <math>x=\text{Re}(z)</math> and the imaginary part <math>y=\text{Im}(z)</math> . The representation of a complex number by its Cartesian coordinates is called the ''Cartesian form'' or ''rectangular form'' or ''algebraic form'' of that complex number.
		+
		+	===Polar form===
		+	Alternatively, the complex number <math>z</math> can be specified by polar coordinates. The polar coordinates are <math>r=\|z\|\ge0</math> , called the '''absolute value''' or '''modulus''', and <math>\phi=\arg(z)</math> , called the '''argument''' of <math>z</math> . For <math>r=0</math> any value of <math>\varphi</math> describes the same number. To get a unique representation, a conventional choice is to set <math>\arg(0)=0</math> . For <math>r>0</math> the argument <math>\varphi</math> is unique modulo <math>2\pi</math> ; that is, if any two values of the complex argument differ by an exact integer multiple of <math>2\pi</math> , they are considered equivalent. To get a unique representation, a conventional choice is to limit <math>\varphi</math> to the interval <math>(-\pi,\pi]</math> i.e. <math>-\pi<\varphi\le\pi</math> . The representation of a complex number by its polar coordinates is called the ''polar form'' of the complex number.
		+
		+	===Conversion from the polar form to the Cartesian form===
		+	:<math>x=r\cos(\varphi)</math>
		+	:<math>y=r\sin(\varphi)</math>
		+
		+	===Conversion from the Cartesian form to the polar form===
		+	:<math>r=\sqrt{x^2+y^2}</math>
		+
		+	:<math>\varphi=
		+	\begin{cases}
		+	\arctan\left(\frac{y}{x}\right)&\text{if }x>0\\
		+	\arctan\left(\frac{y}{x}\right)+\pi&\text{if }x<0,y\ge0\\
		+	\arctan\left(\frac{y}{x}\right)-\pi&\text{if }x<0,y<0\\
		+	\frac{\pi}{2}&\text{if }x=0,y>0\\
		+	-\frac{\pi}{2}&\text{if }x=0,y<0\\
		+	\text{undefined}&\text{if }x=0,y=0
		+	\end{cases}</math>
		+
		+	The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function. A formula that uses the arccos function requires fewer case differentiations:
		+	:<math>\varphi=
		+	\begin{cases}
		+	\arccos\left(\frac{x}{r}\right)&\text{if }y\ge0,r\ne0\\
		+	-\arccos\left(\frac{x}{r}\right)&\text{if }y<0\\
		+	\text{undefined}&\text{if }r=0
		+	\end{cases}</math>
		+
		+	===Notation of the polar form===
		+	The notation of the polar form as
		+	:<math>z=r\big(\cos(\varphi)+i\sin(\varphi)\big)</math>
		+	is called ''trigonometric form''. The notation <math>\text{cis}(\varphi)</math> is sometimes used as an abbreviation for <math>\cos(\varphi)+i\sin(\varphi)</math> . Using [[w:Euler's formula\|Euler's formula]] it can also be written as
		+	:<math>z=re^{i\varphi}</math>
		+	which is called ''exponential form''.
		+
		+	===Multiplication, division, exponentiation, and root extraction in the polar form===
		+	Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.
		+
		+	Using [[w:List of trigonometric identities#Angle sum and difference identities\|sum and difference identities]] its possible to obtain that
		+
		+	:<math>r_1e^{i\varphi_1}\cdot r_2e^{i\varphi_2}=r_1r_2e^{i(\varphi_1+\varphi_2)}</math>
		+
		+	and that
		+
		+	:<math>\frac{r_1e^{i\varphi_1}}{r_2e^{i\varphi_2}}=\frac{r_1}{r_2}\cdot e^{i(\varphi_1-\varphi_2)}</math>
		+
		+	Exponentiation with integer exponents; according to [[w:De Moivre's formula\|de Moivre's formula]],
		+
		+	:<math>\big(re^{i\varphi}\big)^n=r^ne^{ni\varphi}</math>
		+
		+	Exponentiation with arbitrary complex exponents is discussed in the article on [[exponentiation]].
		+
		+	The addition of two complex numbers is just the addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.
		+
		+	Multiplication by <math>i</math> corresponds to a counter-clockwise rotation by 90° or <math>\frac{\pi}{2}</math> radians. The geometric content of the equation <math>i^2=-1</math> is that a sequence of two 90° rotations results in a 180° (<math>\pi</math> radians) rotation. Even the fact <math>(-1)\cdot(-1)=1</math> from arithmetic can be understood geometrically as the combination of two 180° turns.
		+
		+	All the roots of any number, real or complex, may be found with a simple algorithm. The <math>n</math>-th roots are given by
		+
		+	:<math>\sqrt[n]{re^{i\varphi}}=\sqrt[n]{r}\,e^{i\left(\frac{\varphi+2k\pi}{n}\right)}</math>
		+
		+	for <math>k=0,1,2,\ldots,n-1</math> , where <math>\sqrt[n]{r}</math> represents the principal <math>n</math>-th root of <math>r</math> .
		+
		+	==Absolute value, conjugation and distance==
		+	The '''absolute value''' (or ''modulus'' or ''magnitude'') of a complex number <math>z=re^{i\varphi}</math> is defined as <math>\|z\|=r</math> .
		+
		+	Algebraically, if <math>z=a+bi</math> then <math>\|z\|=\sqrt{a^2+b^2}</math> .<!--keep sentence-terminator within math element to make it align better with the formula-->
		+
		+	One can check readily that the absolute value has three important properties:
		+
		+	:<math>\|z\|=0</math> if and only if <math>z=0</math>
		+	:<math>\|z+w\|\le\|z\|+\|w\|</math> (triangle inequality)
		+	:<math>\|z\cdot w\|=\|z\|\cdot\|w\|</math>
		+
		+	for all complex numbers <math>z,w</math> . It then follows, for example, that <math>\|1\|=1</math> and <math>\left\|\frac{z}{w}\right\|=\frac{\|z\|}{\|w\|}</math> . By defining the '''distance''' function <math>d(z,w)=\|z-w\|</math> we turn the set of complex numbers into a [[w:metric space\|metric space]] and we can therefore talk about limits and continuity.
		+
		+	The '''complex conjugate''' of the complex number <math>z=a+bi</math> is defined to be <math>a-bi</math> , written as <math>\bar z</math> or <math>z^*</math> . As seen in the figure, <math>\bar z</math> is the "reflection" of <math>z</math> about the real axis. The following can be checked:
		+	:<math>\overline{z+w}=\bar z+\bar w</math>
		+	:<math>\overline{z\cdot w}=\bar z\cdot\bar w</math>
		+	:<math>\overline{\left(\frac{z}{w}\right)}=\frac{\bar z}{\bar w}</math>
		+	:<math>\bar{\bar z}=z</math>
		+	:<math>\bar z=z</math> if and only if <math>z</math> is real
		+	:<math>\|z\|=\|\bar z\|</math>
		+	:<math>\|z\|^2=z\cdot\bar z</math>
		+	:<math>z^{-1}=\bar z\cdot\|z\|^{-2}</math> if <math>z</math> is non-zero.
		+
		+	The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
		+
		+	That conjugation commutes with all the algebraic operations (and many functions; ''e.g.'' <math>\sin(\bar z)=\overline{\sin(z)}</math>) is rooted in the ambiguity in choice of <math>i</math> (−1 has two square roots). It is important to note, however, that the function <math>f(z)=\bar z</math> is not complex-differentiable.
		+
		+
		+	==Resources==
		+	* [https://en.wikibooks.org/wiki/Calculus/Complex_numbers Complex Numbers], Wikibooks: Calculus
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/Esparza%201093%20Notes%20Week%2012.pdf The complex plane]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/Esparza%201093%20Notes%20Week%2012.pdf The complex plane]. Written notes created by Professor Esparza, UTSA.

← Older revision		Revision as of 22:28, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/ The complex plane]. Written notes created by Professor Esparza, UTSA.	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/The%20complex%20plane/Esparza%201093%20Notes%20Week%2012.pdf The complex plane]. Written notes created by Professor Esparza, UTSA.