Eilera formula , kas nosaukta Leonarda Eilera vārdā, ir formula komplekso skaitļu analīzē, kas saista trigonometriskās funkcijas ar kompleksas pakāpes funkciju . Eilera formula apgalvo, ka jebkuram reālam skaitlim
x
{\displaystyle x}
Kompleksās plaknes vienības riņķis, kur
y
{\displaystyle y}
x
{\displaystyle x}
e
i
x
=
cos
x
+
i
sin
x
{\displaystyle e^{ix}=\cos {x}+i\sin {x}}
e
{\displaystyle e}
Eilera skaitlis ,
i
{\displaystyle i}
imaginārā vienība un
cos
{\displaystyle \cos }
sin
{\displaystyle \sin }
kosinuss un sinuss .[ 1]
x
{\displaystyle x}
[ 2]
Kad
x
=
π
{\displaystyle x=\pi }
e
i
π
=
−
1
{\displaystyle e^{i\pi }=-1}
Pakāpes funkcija
f
(
z
)
=
e
z
{\displaystyle f(z)=e^{z}}
z
{\displaystyle z}
atvasināma funkcija ar kompleksu mainīgo, kurai izpildās šādas īpašības:
d
f
d
z
=
f
{\displaystyle {\frac {df}{dz}}=f}
f
(
0
)
=
1
{\displaystyle f(0)=1}
Priekš kompleksiem mainīgajiem
z
{\displaystyle z}
Teilora rinda :
e
z
=
1
+
z
1
!
+
z
2
2
!
+
z
3
3
!
+
.
.
.
=
∑
n
=
0
∞
z
n
n
!
{\displaystyle e^{z}=1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+...=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}
Priekš kompleksiem mainīgajiem
z
{\displaystyle z}
robeža :
e
z
=
lim
n
→
∞
(
1
+
z
n
)
n
{\displaystyle e^{z}=\lim _{n\rightarrow \infty }(1+{\frac {z}{n}})^{n}}
Robežās definīcijas izmantotie
n
{\displaystyle n}
veselie skaitļi .
Pastāv vairāki formulas pierādījumi.
Apskatīsim funkciju
f
(
θ
)
=
cos
θ
+
i
sin
θ
e
i
θ
{\displaystyle f(\theta )={\frac {\cos {\theta }+i\sin {\theta }}{e^{i\theta }}}}
f
(
θ
)
=
e
−
i
θ
(
cos
θ
+
i
sin
θ
)
{\displaystyle f(\theta )=e^{-i\theta }(\cos {\theta }+i\sin {\theta })}
θ
{\displaystyle \theta }
θ
{\displaystyle \theta }
d
f
(
θ
)
d
θ
=
−
i
e
−
i
θ
(
cos
θ
+
i
sin
θ
)
+
e
−
i
θ
(
−
sin
θ
+
i
cos
θ
)
)
{\displaystyle {\frac {df(\theta )}{d\theta }}=-ie^{-i\theta }(\cos {\theta }+i\sin {\theta })+e^{-i\theta }(-\sin {\theta }+i\cos {\theta }))}
e
−
i
θ
{\displaystyle e^{-i\theta }}
e
−
i
θ
(
−
i
cos
θ
+
sin
θ
−
sin
θ
+
i
cos
θ
)
{\displaystyle e^{-i\theta }(-i\cos {\theta }+\sin {\theta }-\sin {\theta }+i\cos {\theta })}
d
f
(
θ
)
d
θ
{\displaystyle {\frac {df(\theta )}{d\theta }}}
f
(
θ
)
{\displaystyle f(\theta )}
f
(
0
)
=
1
{\displaystyle f(0)=1}
θ
{\displaystyle \theta }
f
(
θ
)
=
cos
θ
+
i
sin
θ
e
i
θ
=
1
{\displaystyle f(\theta )={\frac {\cos {\theta }+i\sin {\theta }}{e^{i\theta }}}=1}
cos
θ
+
i
sin
θ
=
e
i
θ
{\displaystyle \cos {\theta }+i\sin {\theta }=e^{i\theta }}
Teilora rindas pierādījuma animācija Izmantojot imaginārās vienības
i
{\displaystyle i}
i
0
=
1
,
i
1
=
i
,
i
2
=
−
1
,
i
3
=
−
i
,
i
4
=
1
,
i
5
=
i
,
i
6
=
−
1
,
i
7
=
−
i
⋮
⋮
⋮
⋮
{\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}}}
Tagad var izmantot kompleksās pakāpes Teilora rindas definīciju priekš reāliem skaitļiem
x
{\displaystyle x}
e
i
x
=
1
+
i
x
+
(
i
x
)
2
2
!
+
(
i
x
)
3
3
!
+
(
i
x
)
4
4
!
+
(
i
x
)
5
5
!
+
(
i
x
)
6
6
!
+
(
i
x
)
7
7
!
+
(
i
x
)
8
8
!
+
⋯
=
1
+
i
x
−
x
2
2
!
−
i
x
3
3
!
+
x
4
4
!
+
i
x
5
5
!
−
x
6
6
!
−
i
x
7
7
!
+
x
8
8
!
+
⋯
=
(
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
x
8
8
!
−
⋯
)
+
i
(
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
+
⋯
)
=
cos
x
+
i
sin
x
,
{\displaystyle {\begin{aligned}e^{ix}&=1+ix+{\frac {(ix)^{2}}{2!}}+{\frac {(ix)^{3}}{3!}}+{\frac {(ix)^{4}}{4!}}+{\frac {(ix)^{5}}{5!}}+{\frac {(ix)^{6}}{6!}}+{\frac {(ix)^{7}}{7!}}+{\frac {(ix)^{8}}{8!}}+\cdots \\[8pt]&=1+ix-{\frac {x^{2}}{2!}}-{\frac {ix^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {ix^{5}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {ix^{7}}{7!}}+{\frac {x^{8}}{8!}}+\cdots \\[8pt]&=\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}-\cdots \right)+i\left(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \right)\\[8pt]&=\cos x+i\sin x,\end{aligned}}}
Pēdējā solī tiek pamanīts, ka divas bezgalīgās rindas konverģē uz
cos
x
{\displaystyle \cos {x}}
sin
x
{\displaystyle \sin {x}}
absolūtās vērtības konverģē.
Saistība sinusa, kosinusa un imaginārās pakāpes funkcijai. Izmantojot Eilera formulu, trigonometrisko funkciju definīcijas un pakāpju īpašības, iespējams pierādīt lielu daļu trigonometrisko identitāšu.
cos
x
=
Re
(
e
i
x
)
=
e
i
x
+
e
−
i
x
2
,
sin
x
=
Im
(
e
i
x
)
=
e
i
x
−
e
−
i
x
2
i
.
{\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}}}
[ 2]
Šīs divas izteiksmes var iegūt, saskaitot un atņemot Eilera formulas un izsakot
cos
{\displaystyle \cos }
sin
{\displaystyle \sin }
e
i
x
=
cos
x
+
i
sin
x
,
e
−
i
x
=
cos
(
−
x
)
+
i
sin
(
−
x
)
=
cos
x
−
i
sin
x
{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x,\\e^{-ix}&=\cos(-x)+i\sin(-x)=\cos x-i\sin x\end{aligned}}}
Kompleksās pakāpes var vienkāršot trigonometriju, jo tās ir vieglāk pārveidot. Piemēram:
cos
x
cos
y
=
e
i
x
+
e
−
i
x
2
⋅
e
i
y
+
e
−
i
y
2
=
1
2
⋅
e
i
(
x
+
y
)
+
e
i
(
x
−
y
)
+
e
i
(
−
x
+
y
)
+
e
i
(
−
x
−
y
)
2
=
1
2
(
e
i
(
x
+
y
)
+
e
−
i
(
x
+
y
)
2
+
e
i
(
x
−
y
)
+
e
−
i
(
x
−
y
)
2
)
=
1
2
(
cos
(
x
+
y
)
+
cos
(
x
−
y
)
)
.
{\displaystyle {\begin{aligned}\cos x\cos y&={\frac {e^{ix}+e^{-ix}}{2}}\cdot {\frac {e^{iy}+e^{-iy}}{2}}\\&={\frac {1}{2}}\cdot {\frac {e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2}}\\&={\frac {1}{2}}{\bigg (}{\frac {e^{i(x+y)}+e^{-i(x+y)}}{2}}+{\frac {e^{i(x-y)}+e^{-i(x-y)}}{2}}{\bigg )}\\&={\frac {1}{2}}\left(\cos(x+y)+\cos(x-y)\right).\end{aligned}}}
![{\displaystyle {\begin{aligned}e^{ix}&=1+ix+{\frac {(ix)^{2}}{2!}}+{\frac {(ix)^{3}}{3!}}+{\frac {(ix)^{4}}{4!}}+{\frac {(ix)^{5}}{5!}}+{\frac {(ix)^{6}}{6!}}+{\frac {(ix)^{7}}{7!}}+{\frac {(ix)^{8}}{8!}}+\cdots \\[8pt]&=1+ix-{\frac {x^{2}}{2!}}-{\frac {ix^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {ix^{5}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {ix^{7}}{7!}}+{\frac {x^{8}}{8!}}+\cdots \\[8pt]&=\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}-\cdots \right)+i\left(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \right)\\[8pt]&=\cos x+i\sin x,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/704373ea51b842453c839ec11f3f54c1022cf586)
