DeMoivere’s Theorem - Revision history

Khanh: /* Proof by induction */

2022-01-15T19:41:08Z

Proof by induction

Khanh: /* Proof by induction */

2022-01-15T19:37:51Z

Proof by induction

Khanh at 23:12, 28 October 2021

2021-10-28T23:12:58Z

Khanh at 06:07, 9 October 2021

2021-10-09T06:07:55Z

Khanh at 06:04, 9 October 2021

2021-10-09T06:04:11Z

Show changes

Lila at 02:09, 24 September 2021

2021-09-24T02:09:20Z

Rylee.taylor: fixing link to go straight to pdf

2020-08-12T22:29:21Z

fixing link to go straight to pdf

Johnraymond.yanez at 13:11, 12 August 2020

2020-08-12T13:11:10Z

Rylee.taylor: added DeMoivere’s Theorem notes by professor Esparza

2020-08-11T22:05:09Z

added DeMoivere’s Theorem notes by professor Esparza

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/ DeMoivere’s Theorem]. Written notes created by Professor Esparza, UTSA.

@@ Line 49: / Line 49: @@
 See angle sum and difference identities.
-We deduce that {{math|S(''k'')}} implies {{math|S(''k'' + 1)}}. By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, {{math|S(0)}} is clearly true since <math> cos(0x) + i sin(0x) = 1 + 0i = 1</math>. Finally, for the negative integer cases, we consider an exponent of {{math|−''n''}} for natural {{mvar|n}}.
+We deduce that {{math|S(''k'')}} implies {{math|S(''k'' + 1)}}. By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, {{math|S(0)}} is clearly true since <math> \cos(0x) + i \sin(0x) = 1 + 0i = 1</math>. Finally, for the negative integer cases, we consider an exponent of {{math|−''n''}} for natural {{mvar|n}}.
 :<math>\begin{align}
@@ Line 58: / Line 58: @@
 The equation (*) is a result of the identity
 :<math>z^{-1} = \frac{\bar z}{|z|^2},</math>
-for <math> z = cos nx + i sin nx </math>. Hence, {{math|S(''n'')}} holds for all integers {{mvar|n}}.
+for <math> z = \cos nx + i \sin nx </math>. Hence, {{math|S(''n'')}} holds for all integers {{mvar|n}}.
 == Formulae for cosine and sine individually ==

@@ Line 49: / Line 49: @@
 See angle sum and difference identities.
-We deduce that {{math|S(''k'')}} implies {{math|S(''k'' + 1)}}. By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, {{math|S(0)}} is clearly true since {{math|cos(0''x'') + ''i'' sin(0''x'') {{=}} 1 + 0''i'' {{=}} 1}}. Finally, for the negative integer cases, we consider an exponent of {{math|−''n''}} for natural {{mvar|n}}.
+We deduce that {{math|S(''k'')}} implies {{math|S(''k'' + 1)}}. By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, {{math|S(0)}} is clearly true since <math> cos(0x) + i sin(0x) = 1 + 0i = 1</math>. Finally, for the negative integer cases, we consider an exponent of {{math|−''n''}} for natural {{mvar|n}}.
 :<math>\begin{align}
@@ Line 58: / Line 58: @@
 The equation (*) is a result of the identity
 :<math>z^{-1} = \frac{\bar z}{|z|^2},</math>
-for {{math|''z'' {{=}} cos ''nx'' + ''i'' sin ''nx''}}. Hence, {{math|S(''n'')}} holds for all integers {{mvar|n}}.
+for <math> z = cos nx + i sin nx </math>. Hence, {{math|S(''n'')}} holds for all integers {{mvar|n}}.
 == Formulae for cosine and sine individually ==

@@ Line 167: / Line 167: @@
 * [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Trigonometry_(Sundstrom_and_Schlicker)/05%3A_Complex_Numbers_and_Polar_Coordinates/5.03%3A_DeMoivres_Theorem_and_Powers_of_Complex_Numbers De Moivre's Theorem and Powers of Complex Numbers], Mathematics LibreTexts
-==References==
+== Licensing ==
-#Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of Mathematical Functions. New York: Dover Publications. p. 74. ISBN 0-486-61272-4..
+Content obtained and/or adapted from:
-#Lial, Margaret L.; Hornsby, John; Schneider, David I.; Callie J., Daniels (2008). College Algebra and Trigonometry (4th ed.). Boston: Pearson/Addison Wesley. p. 792. ISBN 9780321497444.
+* [https://en.wikipedia.org/wiki/De_Moivre%27s_formula De Moivre's formula, Wikipedia] under a CC BY-SA license

@@ Line 117: / Line 117: @@
 where {{mvar|k}} varies over the integer values from 0 to {{math|''n'' − 1}}.
-This formula is also sometimes known as de Moivre's formula.<ref>{{springer|title=De Moivre formula|id=p/d030300}}</ref>
+This formula is also sometimes known as de Moivre's formula.
 == Analogues in other settings ==

← Older revision		Revision as of 02:09, 24 September 2021
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/Esparza%201093%20Notes%20Week%2012.pdf ~~DeMoivere’s~~ Theorem]. Written notes created by Professor Esparza, UTSA.	+	De Moivre's Theorem states that for a complex number <math> z = r(\cos{\theta} + i\sin{\theta}) </math> and integer <math> n </math>, then
−	* [https://www.youtube.com/watch?v=hTKXSIT_MpU Application of ~~DeMoivere~~'s Theorem]. Produced by Professor Zachary Sharon, UTSA.	+
		+	<math> [r(\cos{\theta} + i\sin{\theta})]^n = r^n(\cos{(n\theta)} + i\sin{(n\theta}))</math>.
		+
		+	==Resources==
		+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/Esparza%201093%20Notes%20Week%2012.pdf De Moivre’s Theorem]. Written notes created by Professor Esparza, UTSA.
		+	* [https://www.youtube.com/watch?v=hTKXSIT_MpU Application of DeMoivre's Theorem]. Produced by Professor Zachary Sharon, UTSA.
		+	* [https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Trigonometry_(Sundstrom_and_Schlicker)/05%3A_Complex_Numbers_and_Polar_Coordinates/5.03%3A_DeMoivres_Theorem_and_Powers_of_Complex_Numbers De Moivre's Theorem and Powers of Complex Numbers], Mathematics LibreTexts

← Older revision		Revision as of 22:29, 12 August 2020
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−	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/ DeMoivere’s Theorem]. Written notes created by Professor Esparza, UTSA.	+	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/Esparza%201093%20Notes%20Week%2012.pdf DeMoivere’s Theorem]. Written notes created by Professor Esparza, UTSA.
	* [https://www.youtube.com/watch?v=hTKXSIT_MpU Application of DeMoivere's Theorem]. Produced by Professor Zachary Sharon, UTSA.		* [https://www.youtube.com/watch?v=hTKXSIT_MpU Application of DeMoivere's Theorem]. Produced by Professor Zachary Sharon, UTSA.

← Older revision		Revision as of 13:11, 12 August 2020
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	* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/ DeMoivere’s Theorem]. Written notes created by Professor Esparza, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/DeMoivere%E2%80%99s%20Theorem/ DeMoivere’s Theorem]. Written notes created by Professor Esparza, UTSA.
		+	* [https://www.youtube.com/watch?v=hTKXSIT_MpU Application of DeMoivere's Theorem]. Produced by Professor Zachary Sharon, UTSA.