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Display information for equation id:math.219374.44 on revision:219374

* Page found: Euler's formula (eq math.219374.44)

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Hash: b6a0a01f9e7d599ed9bb5021a6b67a53

TeX (original user input):

i e ^{ix}  = (\cos(\theta) + i \sin(\theta)) \frac{dr}{dx} + r (-\sin(\theta) + i \cos(\theta)) \frac{d \theta}{dx}\,.

TeX (checked):

ie^{ix}=(\cos(\theta )+i\sin(\theta )){\frac {dr}{dx}}+r(-\sin(\theta )+i\cos(\theta )){\frac {d\theta }{dx}}\,.

LaTeXML (experimental; uses MathML) rendering

MathML (13.565 KB / 1.726 KB) :

i e i x = ( cos ( θ ) + i sin ( θ ) ) d r d x + r ( - sin ( θ ) + i cos ( θ ) ) d θ d x . 𝑖 superscript 𝑒 𝑖 𝑥 𝜃 𝑖 𝜃 𝑑 𝑟 𝑑 𝑥 𝑟 𝜃 𝑖 𝜃 𝑑 𝜃 𝑑 𝑥 {\displaystyle ie^{{ix}}=(\cos(\theta)+i\sin(\theta)){\frac{dr}{dx}}+r(-\sin(% \theta)+i\cos(\theta)){\frac{d\theta}{dx}}\,.}
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                <times id="p1.1.m1.1.33.2.3.cmml" xref="p1.1.m1.1.33.2.3"/>
                <ci id="p1.1.m1.1.33.2.1.cmml" xref="p1.1.m1.1.33.2.1">𝑑</ci>
                <ci id="p1.1.m1.1.33.2.2.cmml" xref="p1.1.m1.1.33.2.2">𝜃</ci>
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                <ci id="p1.1.m1.1.33.3.1.cmml" xref="p1.1.m1.1.33.3.1">𝑑</ci>
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    </annotation-xml>
    <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle ie^{{ix}}=(\cos(\theta)+i\sin(\theta)){\frac{dr}{dx}}+r(-\sin(%
\theta)+i\cos(\theta)){\frac{d\theta}{dx}}\,.}</annotation>
  </semantics>
</math>

SVG (16.576 KB / 4.563 KB) :

i times e Superscript i times x Baseline equals left-parenthesis cosine left-parenthesis theta right-parenthesis plus i times sine left-parenthesis theta right-parenthesis right-parenthesis times StartFraction d times r Over d times x EndFraction plus r times left-parenthesis minus sine left-parenthesis theta right-parenthesis plus i times cosine left-parenthesis theta right-parenthesis right-parenthesis times StartFraction d times theta Over d times x EndFraction period

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Translations to Computer Algebra Systems

Translation to Maple

In Maple: i*(e)^(i*x)=(cos(theta)+ i*sin(theta))*(d*r)/(d*x)+ r*(- sin(theta)+ i*cos(theta))*(d*theta)/(d*x)

Information about the conversion process:

\cos: Cosine; Example: \cos@@{z}

Will be translated to: cos($0)

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E2

Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos


\sin: Sine; Example: \sin@@{z}

Will be translated to: sin($0)

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E1

Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin


I: You use a typical letter for a constant [the imaginary unit == the principal square root of -1].

We keep it like it is! But you should know that Maple uses I for this constant.

If you want to translate it as a constant, use the corresponding DLMF macro \iunit


e: the mathematical constant e == Napier's constant == 2.71828182845... was translated to: e

exp(1): You use a typical letter for a constant [the mathematical constant e == Napier's constant == 2.71828182845...].

We keep it like it is! But you should know that Maple uses exp(1) for this constant.

If you want to translate it as a constant, use the corresponding DLMF macro \expe


i: the imaginary unit == the principal square root of -1 was translated to: i


Translation to Mathematica

In Mathematica: i*(e)^(i*x)=(Cos[\[Theta]]+ i*Sin[\[Theta]])*Divide[d*r,d*x]+ r*(- Sin[\[Theta]]+ i*Cos[\[Theta]])*Divide[d*\[Theta],d*x]

Information about the conversion process:

\cos: Cosine; Example: \cos@@{z}

Will be translated to: Cos[$0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E2

Mathematica: https://reference.wolfram.com/language/ref/Cos.html


\sin: Sine; Example: \sin@@{z}

Will be translated to: Sin[$0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E1

Mathematica: https://reference.wolfram.com/language/ref/Sin.html


E: You use a typical letter for a constant [the mathematical constant e == Napier's constant == 2.71828182845...].

We keep it like it is! But you should know that Mathematica uses E for this constant.

If you want to translate it as a constant, use the corresponding DLMF macro \expe


I: You use a typical letter for a constant [the imaginary unit == the principal square root of -1].

We keep it like it is! But you should know that Mathematica uses I for this constant.

If you want to translate it as a constant, use the corresponding DLMF macro \iunit


e: the mathematical constant e == Napier's constant == 2.71828182845... was translated to: e

i: the imaginary unit == the principal square root of -1 was translated to: i


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