উইকিপিডিয়া, মুক্ত বিশ্বকোষ থেকে
অয়লারের সূত্র জটিল বিশ্লেষণের একটি গাণিতিক সূত্র যা ত্রিকোণমিতিক ফাংশন এবং জটিল সূচকীয় ফাংশনগুলির মধ্যে মৌলিক সম্পর্ক স্থাপন করে। এ সূত্রটির নামকরণ করা হয় বিখ্যাত গণিতবিদ লিওনার্দ অয়লারের নামানুসারে। এ সূত্রানুসারে যে কোন বাস্তব সংখ্যা
x
{\displaystyle x}
e
i
x
=
cos
x
+
i
sin
x
{\displaystyle {e}^{ix}=\cos {x}+i\sin {x}}
যেখানে
e
{\displaystyle e}
i
{\displaystyle i}
c
o
s
,
s
i
n
{\displaystyle cos,sin}
x
{\displaystyle x}
cos
(
x
)
{\displaystyle \cos(x)}
x
{\displaystyle x}
অয়লারে সূত্রে
x
=
π
{\displaystyle x=\pi }
e
i
π
+
1
=
0
{\displaystyle e^{i\pi }+1=0}
অয়লারের অভেদ নামে পরিচিত।
অয়লারের সূত্র। টেইলরের ধারার সাহায্যে [ সম্পাদনা ]
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]&{}=(1-{\frac {{x}^{2}}{2!}}+{\frac {{x}^{4}}{4!}}-{\frac {{x}^{6}}{6!}}+{\frac {{x}^{8}}{8!}}+\cdots )+i(x-{\frac {{x}^{3}}{3!}}+{\frac {{x}^{5}}{5!}}-{\frac {{x}^{7}}{7!}}+\cdots )\\[8pt]&{}=\cos {x}+i\sin {x}\\[8pt]\end{aligned}}}
ধরা যাক,
f
(
x
)
=
e
−
i
x
(
cos
x
+
i
sin
x
)
{\displaystyle f(x)=e^{-ix}(\cos {x}+i\sin {x})}
f
(
x
)
{\displaystyle f(x)}
x
{\displaystyle x}
অন্তরীকরণ করে,
f
′
(
x
)
=
e
−
i
x
(
−
sin
x
+
i
cos
x
)
−
i
e
−
i
x
(
cos
x
+
i
sin
x
)
=
0
{\displaystyle f'(x)=e^{-ix}(-\sin {x}+i\cos {x})-ie^{-ix}(\cos {x}+i\sin {x})=0}
যেহেতু ফাংশন
f
{\displaystyle f}
f
(
x
)
=
e
−
x
(
cos
x
+
i
sin
x
)
=
c
{\displaystyle f(x)=e^{-x}(\cos {x}+i\sin {x})=c}
c
{\displaystyle c}
x
=
0
{\displaystyle x=0}
f
(
0
)
=
1
=
c
{\displaystyle f(0)=1=c}
সুতরাং,
f
(
x
)
=
e
−
x
(
cos
x
+
i
sin
x
)
=
1
{\displaystyle f(x)=e^{-x}(\cos {x}+i\sin {x})=1}
অর্থাৎ
e
i
x
=
cos
x
+
i
sin
x
{\displaystyle e^{ix}=\cos {x}+i\sin {x}}
ধরা যাক,
z
=
cos
x
+
i
sin
x
∴
d
z
=
(
−
sin
x
+
i
cos
x
)
d
x
d
z
=
(
i
2
sin
x
+
i
cos
x
)
d
x
d
z
=
i
(
cos
x
+
i
sin
x
)
d
x
d
z
=
i
z
d
x
d
z
z
=
i
d
x
∫
d
z
z
=
i
∫
d
x
ln
z
=
i
x
+
c
{\displaystyle {\begin{aligned}z&{}=\cos {x}+i\sin {x}\\[8pt]\therefore dz&{}=(-\sin {x}+i\cos {x})dx\\[8pt]dz&{}=(i^{2}\sin {x}+i\cos {x})dx\\[8pt]dz&{}=i(\cos {x}+i\sin {x})dx\\[8pt]dz&{}=izdx\\[8pt]{\frac {dz}{z}}&{}=idx\\[8pt]\int {\frac {dz}{z}}&{}=i\int {dx}\\[8pt]\ln {z}&{}=ix+c\\\end{aligned}}}
যেখানে
c
{\displaystyle c}
সমাকলন ধ্রুবক। প্রথম লাইনের সমীকরণে
x
=
0
{\displaystyle x=0}
z
=
1
{\displaystyle z=1}
x
=
0
{\displaystyle x=0}
z
=
1
{\displaystyle z=1}
c
=
0
{\displaystyle c=0}
সুতরাং,
ln
z
=
i
x
{\displaystyle \ln {z}=ix}
e
i
x
=
cos
x
+
i
sin
x
{\displaystyle e^{ix}=\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]&{}=(1-{\frac {{x}^{2}}{2!}}+{\frac {{x}^{4}}{4!}}-{\frac {{x}^{6}}{6!}}+{\frac {{x}^{8}}{8!}}+\cdots )+i(x-{\frac {{x}^{3}}{3!}}+{\frac {{x}^{5}}{5!}}-{\frac {{x}^{7}}{7!}}+\cdots )\\[8pt]&{}=\cos {x}+i\sin {x}\\[8pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3536be258b73fb3c58c0477fd3beadf6c50a5d20)
![{\displaystyle {\begin{aligned}z&{}=\cos {x}+i\sin {x}\\[8pt]\therefore dz&{}=(-\sin {x}+i\cos {x})dx\\[8pt]dz&{}=(i^{2}\sin {x}+i\cos {x})dx\\[8pt]dz&{}=i(\cos {x}+i\sin {x})dx\\[8pt]dz&{}=izdx\\[8pt]{\frac {dz}{z}}&{}=idx\\[8pt]\int {\frac {dz}{z}}&{}=i\int {dx}\\[8pt]\ln {z}&{}=ix+c\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59cc49aa8680312438adfb701a27e536a05a0841)
