Revision #1203 → #1822 · back to history
modifiedEuler's formula7faf88527fc9
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| anchors | [{"section":"(Lead)","snippet":"Euler's formula states that, for any real number x"},{"type":"math_alttext","value":"{\\displaystyle e^{ix}=\\cos x+i\\sin x,}"}] | — |
modifiedcis notation9d84057f2cdc
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| anchors | [{"section":"(Lead)","snippet":"This complex exponential function is sometimes denoted cis x"},{"type":"math_alttext","value":"{\\displaystyle e^{ix}=\\cos x+i\\sin x,}"}] | — |
modifiedCotes's logarithmic identity65e9b71a4413
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| anchors | [{"section":"History","snippet":"Roger Cotes presented a geometrical argument that can be interpreted"},{"type":"math_alttext","value":"{\\displaystyle ix=\\ln(\\cos x+i\\sin x).}"}] | — |
modifiedBernoulli's integral identityab493b179336
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| anchors | [{"section":"History","snippet":"Johann Bernoulli had found that"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {1}{1+x^{2}}}={\\frac {1}{2}}\\left({\\frac {1}{1-ix}}+{\\frac {1}{1+ix}}\\right).}"}] | — |
modifiedDifferential equation definition of exp449d0fa007f9
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| anchors | [{"section":"Differential equation definition","snippet":"the unique differentiable function of a complex variable for which the derivative equals the function"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {df}{dz}}=f}"},{"type":"math_alttext","value":"{\\displaystyle f(0)=1.}"}] | — |
modifiedPower series definition of exp37b566d8936b
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| anchors | [{"section":"Power series definition","snippet":"For complex z"},{"type":"math_alttext","value":"{\\displaystyle e^{z}=1+{\\frac {z}{1!}}+{\\frac {z^{2}}{2!}}+{\\frac {z^{3}}{3!}}+\\cdots =\\sum _{n=0}^{\\infty }{\\frac {z^{n}}{n!}}.}"}] | — |
addedExponential function is never zero8e345162fa60
modifiedProof via power seriesfa2f550dcbc1
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| anchors | [{"section":"Using power series","snippet":"Here is a proof of Euler's formula using power-series expansions"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}i^{0}&=1,&i^{1}&=i,&i^{2}&=-1,&i^{3}&=-i,\\\\i^{4}&=1,&i^{5}&=i,&i^{6}&=-1,&i^{7}&=-i\\\\&\\vdots &&\\vdots &&\\vdots &&\\vdots \\end{aligned}}}"}] | — |
modifiedProof via polar coordinates931db3f178e8
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| anchors | [{"section":"Using polar coordinates","snippet":"Another proof"},{"type":"math_alttext","value":"{\\displaystyle e^{ix}=r\\left(\\cos \\theta +i\\sin \\theta \\right).}"},{"type":"math_alttext","value":"{\\displaystyle ie^{ix}=\\left(\\cos \\theta +i\\sin \\theta \\right){\\frac {dr}{dx}}+r\\left(-\\sin \\theta +i\\cos \\theta \\right){\\frac {d\\theta }{dx}}.}"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\theta }=re^{iC}(\\cos \\theta +i\\sin \\theta );}"},{"type":"math_alttext","value":"{\\displaystyle 1=re^{iC}(\\cos(0)+i\\sin(0))=re^{iC}(1+i0)=re^{iC},}"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\theta }=\\cos \\theta +i\\sin \\theta .}"}] | — |
modifiedPolar form of complex numbersd6469a7f21ab
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| anchors | [{"section":"Interpretation of the formula","snippet":"Any complex number z = x + iy , and its complex conjugate"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}z&=x+iy=|z|(\\cos \\varphi +i\\sin \\varphi )=re^{i\\varphi },\\\\{\\bar {z}}&=x-iy=|z|(\\cos \\varphi -i\\sin \\varphi )=re^{-i\\varphi },\\end{aligned}}}"}] | — |
addedArgument of a complex number67a770328620
modifiedComplex logarithm4cfb7281202c
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| anchors | [{"section":"Use of the formula to define the logarithm of complex numbers","snippet":"this can be used as the definition for the complex logarithm"},{"type":"math_alttext","value":"{\\displaystyle a=e^{\\ln a},}"},{"type":"math_alttext","value":"{\\displaystyle e^{a}e^{b}=e^{a+b},}"},{"type":"math_alttext","value":"{\\displaystyle z=\\left|z\\right|e^{i\\varphi }=e^{\\ln \\left|z\\right|}e^{i\\varphi }=e^{\\ln \\left|z\\right|+i\\varphi }}"},{"type":"math_alttext","value":"{\\displaystyle \\ln z=\\ln \\left|z\\right|+i\\varphi ,}"}] | — |
modifiedde Moivre's formula2fa31a06b595
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| anchors | [{"section":"Use of the formula to define the logarithm of complex numbers","snippet":"implies several trigonometric identities , as well as de Moivre's formula"},{"type":"math_alttext","value":"{\\displaystyle \\left(e^{a}\\right)^{k}=e^{ak},}"}] | — |
modifiedSine and cosine as exponentials6738f39c5547
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| anchors | [{"section":"Relationship to trigonometry","snippet":"an interpretation of the sine and cosine functions as weighted sums of the exponential function"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\cos x&=\\operatorname {Re} \\left(e^{ix}\\right)={\\frac {e^{ix}+e^{-ix}}{2}},\\\\\\sin x&=\\operatorname {Im} \\left(e^{ix}\\right)={\\frac {e^{ix}-e^{-ix}}{2i}}.\\end{aligned}}}"}] | — |
modifiedTrig functions for complex argumentsca1321a17e6d
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| anchors | [{"section":"Relationship to trigonometry","snippet":"These formulas can even serve as the definition of the trigonometric functions for complex arguments"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\cos iy&={\\frac {e^{-y}+e^{y}}{2}}=\\cosh y,\\\\\\sin iy&={\\frac {e^{-y}-e^{y}}{2i}}={\\frac {e^{y}-e^{-y}}{2}}i=i\\sinh y.\\end{aligned}}}"}] | — |
addedcos(iy) = cosh y2ab53362d8f3
modifiedQuaternion versor973168e12c10
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| anchors | [{"section":"Other applications","snippet":"For any point r on this sphere, and x a real number, Euler's formula applies"},{"type":"math_alttext","value":"{\\displaystyle \\exp xr=\\cos x+r\\sin x,}"}] | — |
modifiedEuler's formula at x = τ9692ded55fcf
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| anchors | [{"section":"Other special cases","snippet":"The special case at x = τ"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}e^{i\\tau }&=\\cos \\tau +i\\sin \\tau \\\\&=1+0\\end{aligned}}}"}] | — |