“De Moivre's formula”的意思、由来-开放百科全书

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De Moivre's formula

释义

Derivation from Euler's formula
Proof by induction (for integer {{mvar|n}})
Formulae for cosine and sine individually
Failure for non-integer powers, and generalization
Roots of complex numbers
Analogues in other settings
Hyperbolic trigonometry Extension to complex numbers Quaternions Example {{gaps|2|×|2}} matrices
References
External links

{{Use American English|date = March 2019}}{{Short description|1 = Theorem: (cos(x) + i sin(x))^n = cos(nx) + i sin(nx)}}

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any real number {{mvar|x}} and integer {{mvar|n}} it holds that

where {{mvar|i}} is the imaginary unit ({{math|i² {{=}} −1}}). While the formula was named after de Moivre, he never stated it in his works.^[1] The expression {{math|cos(x) + i sin(x)}} is sometimes abbreviated to {{math|cis(x)}}.

The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that {{mvar|x}} is real, it is possible to derive useful expressions for {{math|cos(nx)}} and {{math|sin(nx)}} in terms of {{math|cos(x)}} and {{math|sin(x)}}.

As written, the formula is not valid for non-integer powers {{mvar|n}}. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the {{mvar|n}}th roots of unity, that is, complex numbers {{mvar|z}} such that {{math|zⁿ {{=}} 1}}.

Derivation from Euler's formula

Although historically proven earlier, de Moivre's formula can easily be derived from Euler's formula

and the exponential law for integer powers

Then, by Euler's formula,

Proof by induction (for integer {{mvar|n}})

The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer {{mvar|n}}, call the following statement {{math|S(n)}}:

For {{math|n > 0}}, we proceed by mathematical induction. {{math|S(1)}} is clearly true. For our hypothesis, we assume {{math|S(k)}} is true for some natural {{mvar|k}}. That is, we assume

Now, considering {{math|S(k + 1)}}:

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}}. Finally, for the negative integer cases, we consider an exponent of {{math|−n}} for natural {{mvar|n}}.

The equation (*) is a result of the identity

for {{nowrap|z {{=}} cos (nx) + i sin (nx)}}. Hence, {{math|S(n)}} holds for all integers {{mvar|n}}.

Formulae for cosine and sine individually

Being an equality of complex numbers, one necessarily has equality both of the real parts and of the imaginary parts of both members of the equation. If {{mvar|x}}, and therefore also {{math|cos x}} and {{math|sin x}}, are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century French mathematician François Viète:

In each of these two equations, the final trigonometric function equals one or minus one or zero, thus removing half the entries in each of the sums. These equations are in fact even valid for complex values of {{mvar|x}}, because both sides are entire (that is, holomorphic on the whole complex plane) functions of {{mvar|x}}, and two such functions that coincide on the real axis necessarily coincide everywhere. Here are the concrete instances of these equations for {{math|n {{=}} 2}} and {{math|n {{=}} 3}}:

The right hand side of the formula for {{math|cos nx}} is in fact the value {{math|T_n(cos x)}} of the Chebyshev polynomial {{math|T_n}} at {{math|cos x}}.

Failure for non-integer powers, and generalization

De Moivre's formula does not hold for non-integer powers. The derivation of de Moivre's formula above involves a complex number raised to the integer power {{mvar|n}}. If a complex number is raised to a non-integer power, the result is multiple-valued (see failure of power and logarithm identities). For example, when {{math|n {{=}} {{sfrac|1|2}}}}, de Moivre's formula gives the following results:

for {{math|x {{=}} 0}} the formula gives 1^{{frac|1|2}} = 1, and

for {{math|x {{=}} 2{{pi}}}} the formula gives 1^{{frac|1|2}} = −1.

This assigns two different values for the same expression 1^{{frac|1|2}}, so the formula is not consistent in this case.

On the other hand, the values 1 and −1 are both square roots of 1. More generally, if {{mvar|z}} and {{mvar|w}} are complex numbers, then

is multi-valued while

is not. However, it is always the case that

is one value of

Roots of complex numbers

A modest extension of the version of de Moivre's formula given in this article can be used to find the {{mvar|n}}th roots of a complex number (equivalently, the power of {{math|{{sfrac|1|n}}}}).

If {{mvar|z}} is a complex number, written in polar form as

then the {{mvar|n}} {{mvar|n}}th roots of {{mvar|z}} are given by

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.^[2]

Analogues in other settings

Hyperbolic trigonometry

Since {{math|cosh x + sinh x {{=}} e^x}}, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all {{math|n ∈ ℤ}},

Also, if {{math|n ∈ ℚ}}, then one value of {{math|(cosh x + sinh x)ⁿ}} will be {{math|cosh nx + sinh nx}}.^[3]

Extension to complex numbers

The formula holds for any complex number

where

Quaternions

To find the roots of a quaternion there is an analogous form of de Moivre's formula. A quaternion in the form

can be represented in the form

In this representation,

and the trigonometric functions are defined as

In the case that {{math|a² + b² + c² ≠ 0}},

that is, the unit vector. This leads to the variation of De Moivre's formula:

^[4]

Example

To find the cube roots of

write the quaternion in the form

Then the cube roots are given by:

{{gaps|2|×|2}} matrices

Consider the following matrix

. Then . This fact (although it can be proven in the very same way as for complex numbers) is a direct consequence of the fact that the space of matrices of type is isomorphic to the space of complex numbers.

References

{{cite book|first=Milton |last=Abramowitz |first2=Irene A. |last2=Stegun |title=Handbook of Mathematical Functions |year=1964 |publisher=Dover Publications |location=New York |isbn=0-486-61272-4 |page=74 }}.

1. ^{{cite book |first1=Margaret L. |last1=Lial |first2=John |last2=Hornsby |first3=David I. |last3=Schneider |first4=Daniels |last4=Callie J. |title=College Algebra and Trigonometry|edition=4th |year=2008 |location=Boston |publisher=Pearson/Addison Wesley |isbn=9780321497444 |page=792}}
2. ^{{springer|title=De Moivre formula|id=p/d030300}}
3. ^{{cite journal|last=Mukhopadhyay|first=Utpal|title=Some interesting features of hyperbolic functions|journal=Resonance|date=August 2006|volume=11|issue=8|pages=81–85|doi=10.1007/BF02855783}}
4. ^{{cite journal|last=Brand|first=Louis|title=The roots of a quaternion|journal=The American Mathematical Monthly|date=October 1942|volume=49|issue=8|pages=519–520|jstor=2302858|doi=10.2307/2302858}}

External links

De Moivre's Theorem for Trig Identities by Michael Croucher, Wolfram Demonstrations Project.

2 : Theorems in complex analysis|Articles containing proofs

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