「~Webで連載~ 指数関数と三角関数のつながり」 the relationship of "exp...
TRANSCRIPT
-
2016.01.30
i = 1 z =zRe + i zIm
exp` azRe+i zIm =expzRe` acos zIm+isin zIm
201510http://www008.upp.so-net.ne.jp/web-synapse/WS_ExpCosSin_index.html
@Cello_221Hertz
-
I-1
-
0
I- 1.1
0
0
I-1.1
-
I- 1.2
0
-
x y
x+yA
B
C
I- 1.3
x y
j jx +y j jx +j jy
I-1.3
-
aaa a
I- 1.4
a >0
j jx a < x a
a
a >0
0
-
a
a
X
a
X
I- 1.5
a X a
X a X
I-1.5
-
I- 1.6
x a xa
x a
a x a
xa
-
I-2
-
-2
-1
0
1
2
3
4
5
6
-3 -2 -1 0 1 2 3 4 5
x
F (x )
I- 2.1
x F (x )
F (x ) x
F (x ) (x ) F (x ) x
x F (x )
F (x ) x x
F (x ) F (x )
I-2.1
F (x ) = 21 x 2
21 x1 2< x 4
-
-2
-1
0
1
2
3
4
5
6
-3 -2 -1 0 1 2 3 4 5
x
F (x )
I- 2.2
x x0
F (x ) >0 j jF (x ) F0
-
I- 2.3
A (x ) B (x )
limxx
0
A (x )= A0 limxx
0
B (x )= B0
c
limxx
0
d eA (x )B (x ) = limxx
0
A (x ) limxx
0
B (x )
limxx
0
d ecA (x ) = c limxx
0
A (x )
limxx
0
d eA (x )B (x ) = d elimxx0 A (x ) d elimxx0 B (x )
limxx
0B (x )A (x )
= limxx
0
B (x )
limxx
0
A (x ) lim
xx0
B (x )0
-
I- 2.4
limx0 cos x
log10` ax+1
limxd e3 ` a51
x
limx f gcos x + sin` a2
j jx
1
limx1 x 21
x 2+x 2
-
Ox
x0
F (x0)
x0+x
F (x0+x )
I- 2.5
F (x ) x=x0 x=x0+x
F (x0) F (x0+x )
x=` ax0+x x0 F =F` ax0+x F (x0)
xF =
` ax0+x x0
F` ax0+x F (x0) =x
F` ax0+x F (x0)
x 0 F (x ) x=x0
x=x0
F (x ) x
f gdxdF (x )x=x
0
= limx0 x
F (x0+x )F (x0) ()
I-2.5
-
x0
F (x ) dx
dF (x )
x0 x
dx
dF (x ) = limx0 x
F (x+x ) F (x )
() f gdxdF (x )x=x
0
dx
dF (x ) x=x0
-
I- 2.6
F (x ) x=x0 I-2.5
F (x ) x=x0
-
I- 2.7
F (x )
dx
dF (x ) = dxd F (x ) = F' (x )
I-2.5
xF (x+x ) F (x )
=F'(x ) x0
F (x )=F'(x )x x0
x0 d
dF (x )=F'(x )dx
F (x ) x
-
I- 2.8
A (x ) B (x )
A'(x ) B'(x ) c
dxdd eA (x )B (x ) = A'(x ) B'(x )
dxdd ecA (x ) = cA'(x )
dxdd eA (x )B (x ) = A'(x )B (x ) + A (x )B'(x )
dxdd eA` aB (x ) = A' aB (x ) B'(x )
-
A (x )
B (x )
x A'(x )A (x+x )
B (x+x )
x B' (x )
I- 2.9
dxdd eA (x )B (x ) = A' (x )B (x )+A (x )B'(x )
I-2.9
dxdd eA (x )B (x ) =A' (x )B (x )+A (x )B'(x )
dxdd eA (x )B (x ) A (x +x )B (x+x ) A (x )B (x )
x x
A (x+x )B (x+x ) A (x )B (x )x0
xA'(x )B (x ) xA (x )B' (x )
x 2A' (x )B'(x ) x x0
dxdd eA (x )B (x ) = A' (x )B (x )+A (x )B'(x )
-
xO
B
OB =B (x )
A` aB =` aB (x )
B (x )
x
B
A
I- 2.10
dxdd eA` aB (x ) = A'` aB (x ) B' (x )
I-2.10
A` aB (x ) BA B (x )
xB A x
xA x0
-
limx0 x
A = limx0` aBA xB
limx0 B
A A (B )
A'(B ) limx0 x
B
B (x ) B'(x )
dxdd eA` aB (x ) = A'` aB (x ) B' (x )
-
I- 2.11
n
F (x ) n n 2
F (x ) n1
dx
dF (x ) =dxd F (x ) =F (1)(x ) =F' (x )
dx2
d2F (x ) =dx2d2 F (x ) =
dxd
` adxdF (x ) =F (2)(x ) =F''(x )
n dxn
dnF (x )=
dxndn F (x ) =
dxd
` adx n1dn1F (x )
=F (n)(x )
-
O Ox x
F (x ) F (x )F (x+x )
F (x+x )
x+xx+x
F (x +2x )F (x +2x )
x+2xx+2x
2F (x +2x )+F (x )
2F (x +2x )+F (x )
I- 2.12
F (x )
I-2.12
-
I-3 xn n
-
I- 3.1
a b n
` aa +bn =
r =0
n
nCr a nr b r
` aa +bn=` aa+b ` aa+b ` aa +b
` aa+b n b r
nCr a n r b r
` aa+bn
nCr =r !` an r !
n ! n ! n
1 n !=n` an 1 ` an 2 1
0!=1 nC0 =1
-
I- 3.2
` aa +b5
-
I- 3.3
xn n
dxd x n =nx n 1 n
-
I- 3.4
xn n
F (x )
F (x )=x 5 +2x
F (x )=` a7x 2+14
F (x )=` aax 3+b ` acx 5+d3
a, b, c, d
-
t [s]1O
2g
Y (t ) [m]
I- 3.5
t=0 [s]
t [s] Y (t) [m] g [m/s2]
( g>0 )
Y (t)=21 gt 2 [m]
t [s] Y (t) [m]
Y'(t) [m/s]
Y'(t)
t [s] Y (t) [m]
Y''(t ) [m/s2]
Y''(t )
sSecond
I-3.5 Y (t )=21 gt 2 [m]
-
I-4
-
I- 4.1
a x a x
p q a
a p+q =a pa q
a p q =a p a q
` aa p q =a pq
-
x1
1
a
F (x )
x1
1a
F (x )
0< a < 1 1< a
I- 4.2
a
F (x )=a x 0
-
I- 4.3
A0 [g] T [] t []
A (t) [g]
A (t )= A0` a21Tt
[g]
T =1600 []
A0=1 [g]
A (t) [g]
800 [] [g]
-
xa
a1
1
y
y=FINV(x )
y= xy=F (x )= ax
I- 4.4
F (x ) x
y= F (x )
x y
x =FINV(y ) x y
y =FINV(x )
x FINV(x )
FINV(x ) F (x )
x FINV(x )
I-4.4 F (x )= a x FINV(x )
-
I- 4.5
y a 1 p =a y y
y = loga p
y a p p
p=a y p
r p q a M 1
p =q loga p = loga q
loga` apq = loga p + loga q
loga` aqp = loga p loga q
loga` apr = r loga p
a loga p = p
loga p = logMalogMp
loga p
a= 1 p =1y y
-
I- 4.6
621
log10
2=0.3010 log10
3 = 0.4771
-
O 1 ax
F (x )
1
O 1ax
F (x )
1
0< a < 1 1< a
I- 4.7
a 1
F (x )= loga x 0
-
I- 4.8
A0 [g] T [] t [] At[g]
At =A0` a21Tt
t=
-
I-5
-
I- 5.1
e
e = limn` a1+ n
1 n n
e = limp` a1+ p
1 p p
e=2.1728
n p
e=2.1728
-
I- 5.2
exp x
x exp x
exp x =ex = limn` a1+ nx
n n
exp exp x
exp` a2x
-
I- 5.3
exp x
x
exp x =1+1!x +
2!x 2 +
3!x 3 ++
n !x n + n
exp x I-5.1
n
exp x
-
I- 5.4
exp x
dxd
exp x = exp x
dxd a x =` alogea a x a
-
0
2
2
0
4
4
6
8
-2-4
expx
F (x )= exp xF (x )
x
I- 5.5
exp x
F (x )= exp x exp x
F (x )
-
I- 5.6
y =F (x ) dxdy
x =FINV(y ) dydx
dxdy
dydx
= 1
-
I- 5.7
logex
dxd
logex =
x1
x
dxd
loga x = logea
1x1
a 1
-
I- 5.8
F (x )
F (x )= loge` ax
1 x
F (x )=221
expd e 22` ax2
-
I-6 F(x)=xp p
F(x)=B(x)A(x)
-
I- 6.1
F(x)=xp p
F (x )=x p p
dxd x p =p x p 1
-
I- 6.2
F(x)= x
0
-
I- 6.3
F(x)=B(x)A(x)
A (x ) B (x ) B (x )0
dxd d eB (x )A (x ) = d eB (x ) 2
A'(x )B (x ) A (x )B'(x )
-
II-1
-
II- 1.1
= 3.14159
r 2r
II-1.1
-
: 1
II- 1.2
II-1.2
1
Radian [rad]
[rad]
360 []
deg [] [rad]
=180 deg [rad]
r [rad] l
l = r
-
x
y
O
P
[rad] 0
30
45 60
90 120
135
150
0 [rad]
6 [rad]
4 [rad]
3 [rad]2
[rad]
32 [rad]
43 [rad]
65 [rad]
180
210
225 240 6
7 [rad]
[rad]
45 [rad]
34 [rad]
270
23 [rad]
300 315
330
35 [rad]
47 [rad]
611 [rad]
II- 1.3
P x OP [rad]
II-1.3
II-1.3
-
II-1.3 360 [] 2 [rad] 0 [] 0 [rad]
P +2n [rad]
( n ) P
3
II-1.3 P
0
-
a
bc
II- 1.4
c a b
c 2 =a 2+b 2
a b c c 2 = a 2+b 2
c
II-1.4.1
-
sin
O
P
P'
1
II- 1.5
II-1.5.1
II-1.5.1 O 1
OP [rad]
cos P
sin P
tan =cos sin
cos sin tan 2
-
cos
-
II- 1.6
cos2 + sin2 =1
1 + tan2 =cos2
1
cos2 ` acos2
-
II- 1.7
2
cos` a = cos
sin` a = sin
cos` a 2 = sin sin` a 2 = cos
cos` a = cos
sin` a = sin
+2n n
cosf g +` a2n 21 =sin
-
3
4
5
53.1
36.9
5
12
13
67.4
22.6
1
12
45
45
1
3
2
30
60
II- 1.8
II-1.8
II-1.8
(a)(b)
(c)(d)
II-1.8
(a) (b)
(c) (d)
-
II-1.8
0 22.6 30 36.9 45 53.1 60 67.4
0 0.396p
0.644p
0.933p
1.18
cos 11312
23
54
2
153
21
135
sin 0135
21
53
2
154
23
1312
tan 0125
3
143
134
3 512
90 112.6 120 126.9 135 143.1 150 157.4
2p
1.9732p
2.214
3p2.50
65p
2.75
cos 0 -135
-21
-53 -
2
1-
54
-23 -
1312
sin 11312
23
54
2
153
21
135
tan - -512
- 3 - 34
-1 -43 -
3
1-
125
-
II- 1.9
cos =0.6 sin cos` a
sin` a
-
II-2
-
x
y
O x0
y0
II- 2.1
x
y
II-2.1
-
x
y
O
A
x
y
O
A
A0
A1
II- 2.2
II-2.2.1
A
II-2.2.2
A0 A1 A =A0A1
-
A A0` aA0x , A0y A1` aA1x , A1y
A1 x y
A0 x y
A
A =` aA1x A0x , A1y A0y
A0A1j jA
j jA
j jA = ` aA1x A0y2+` aA1y A0y
2
A A1
A0
1
A =` aAx , Ay
A =j jA
A = ` aAx2+Ay2Ax ,
Ax2+Ay2Ay
-
0 =` a0, 0 0
=
A=` aAx ,Ay B=` aBx ,By
A=B Ax = Bx
Ay =By
-
x
y
O
AA` aAx ,Ay
II- 2.3
II-2.3
-
II- 2.4
A=` aAx , Ay B=` aBx , By k
A+B =B+A =` aAx +Bx , Ay +ByII-2.4.1(a) (b)
kA =` akAx , kAyj jkA =j jk j jA
II-2.4.2
AB =` aAx Bx , Ay ByII-2.4.3(a) (b)
+
-
x
y
O
A
B
A+B
x
y
O
AB A+B
x
y
OA
kA
x
y
O
A
BB
AB
x
y
O
A
B
AB
(a) (b)
II-2.4.1
II-2.4.2
(a) (b)
II-2.4.3
-
x
y
O
A
A` aAx ,Ay
ex
ey
II- 2.5
x y x
ex y ey
ex =` a1,0 ey =` a0,1
ex ey
A` aAx , Ay
OA
OA = Axex +Ayey
II-2.5
-
FW
w [kg]
F1 F2
Fm
II- 2.6
w [kg]
F1 F2
j jF1 =j jF2
g[m/s2]
II-2.6
-
II- 2.7
a b
j ja j jb
A B
j jA j jB
cos
sin
-
x
y
O
A
B
AB
II- 2.8
A=` aAx ,Ay B=` aBx , By A B
AB
A B =j jA j jB cosAB
AB
k C
A B =B A
` akA B = k` aA B
A ` aB+C =A B+A C
A B =Ax Bx + Ay By
II-2.8.1
-
x
y
O
A
BAB
II- 2.9
A=` aAx ,Ay B=` aBx , By A B
AB
A B =j jA j jB sinAB
AB
k C
AB = B A
` akA B = k` aA B
A` aB+C =A B+A C
A B =Ax By Ay Bx
II-2.9.1
-
II- 2.10
F p F=` ap ,F (p )
F (p ) p
A=` aAx , Ay Ax0
F A = S
-
F
x
II- 2.11
F x F
F x
II-2.11
-
II- 2.12
F p F=` ap ,F (p )
F (p ) p
A=` aAx , Ay Ay0
FA =S
-
F
d
O
II- 2.13
O d F
dF
II-2.13
-
II-3
-
A
B Ca
bc
II- 3.1
II-3.1
II-3.1
-
A
B Ca
bc
II- 3.2
II-3.2.1
c 2 =a 2+b 22abcosC
II-3.2.1
a b c
a 2 =b 2+c 22bccosA
b 2 =c 2+a 22cacosB
cosC =2ab
a 2+b 2c 2
cosA =2bc
b 2+c 2a 2
cosB =2ca
c 2+a 2b 2
-
A
B Ca
bc
O
II- 3.3
II-3.3.1 ABC O r
sinA
a =sinB
b =sinC
c = 2r
II-3.3.1
-
II- 3.4
cos` a + = cos cos sin sin
cos` a = cos cos + sin sin
sin` a + = sin cos + cos sin
sin` a = sin cos cos sin
=
cos` a2 = cos2 sin2 =2cos2 1 = 12sin2
sin` a2 =2sin cos
2
cos2` a2 = 21+cos
sin2` a2 = 21 cos
-
II- 3.5
A B
cosA + cosB =2cos` a2A +B cos` a2A B cosA cosB = 2 sin` a2A +B sin` a2A B sinA + sinB =2 sin` a2A +B cos` a2A B sinA sinB =2 cos` a2A +B sin` a2A B
-
II- 3.6
cos1= a
sin1
cos2
sin2
cos3
cos179
sin91
cos3 cos 1sin3+ sin1
-
II- 3.7
1 i
i2 =1 i = 1
17i 3
5i 2i
zRe zIm z
z = zRe +zIm i
zRe Rez
zIm Im z
5+3i 5 3
-
ORez
Im z
z0 = z0Re + z0Im i
z0Re
z0Im z0rz0
II- 3.8
II-3.8.1
II-3.8.1
z0 z0r j jz0
z0r =j jz0 = z0Re2 + z0Im
2
z0 z0 arg z0
z0 = arg z0 = arctan` az0Rez0Im
z0
z0 = z0Re + z0Im i
= z0r` acosz0 + i sin z0
-
z1 z2
Rez1 = Rez2 Im z1 = Im z2
z1 z2
j jz1 =j jz2 argz1 = argz2
z1
z2
-
II- 3.9
m
` acos +i sin m= cos` am +i sin` am
z j jz = zr argz =
z = zr` acos + isin m
z m= zrmf gcos` am +isin` am
-
II-4
-
O S R
R'S'
1
x
II- 4.1
x
sinx x0
limx0 x
sinx=1
II-4.1 limx0 x
sinx=1
-
II- 4.2
cosx sinx
dxd
cosx = sinx
dxd
sinx = cosx
-
II- 4.3
tanx
dxd
tanx =cos2x
1
-
O
L0 [m]
k [N/m] m [kg]
Y0 [m]
II- 4.4
II-4.4
II-4.4 m [kg]
k [N/m] L0 [m]
Y0 [m]
t=0 [s] t [s]
Y (t) [m]
Y (t) [m]
-
II-5
-
dx
x0 x1
II- 5.1
x0 x1 x0 x1
dx x0 x1
x0
x1dx dx x 0
x0
-
x0 x1x
F (x )
ix
x F (x0+ix )
II- 5.2
II-5.2
x x0 dx
x0 x1 n n x0
i i n x0 +ix
x=x0+ix F (x+ ix ) x
x F (x+ix ) II-5.2
x 0 n
x =x0 x=x1 F (x ) x =x0
x= x1
limnf gi =1
n
` ax F (x+ix =x0x1
dxF (x )
-
Fu (x ) F (x ) Fu (x )
F (x )
Fu (x ) F (x )
Fu (x )+c c
F (x ) Fu (x )
x0
x1dxF (x )= Fu (x1)Fu (x0)
F (x ) Fu(x) Fu(x)
F (x )
[ ] |
x0
x1dxF (x )=f gFu (x )
x0
x1= jFu (x )
x0
x1
-
II- 5.3
x1 x
Fu0(x )
Fu0(x )=x0xdxF (x )
Fu0(x ) F (x )
F (x ) dxF (x )
-
II- 5.4
A (x ) B (x ) Au (x ) Bu (x )
A'(x ) B'(x ) x0 x1 c
x0
x1dxd eA (x )B (x ) =x0
x1dxA (x )
x0
x1dxB (x )
=f gAu (x )x0
x1f gBu (x )
x0
x1
x0
x1dxd ecA (x ) =cx0
x1dxA (x )= cf gAu (x )
x0
x1
x0
x1dx A (x )B (x )=f gAu (x )B (x )
x0
x1
x0
x1dx Au (x )B' (x )
=f gA (x )Bu (x )x0
x1
x0
x1dx A'(x )Bu (x )
F (P (t)) x=P (t ) t=PINV(x )
x0=P (t0) t0=PINV(x0)
x1=P (t1) t1=PINV(x1)
x0
x1dxF (x )=
t0
t1dt F (P (t ))P'(t )
P'(t ) P'(t) =dt
dP (t)
-
II- 5.5
dx x p =p+1
1 x p+1 p p 1
dx x1
= logej jx
dx logex =x loge x x
dx exp x = exp x
dx cosx = sinx
dx sinx =cosx
-
II- 5.6
F(x) =xm m
F (x )=x m m
Fu (x ) x 21
x1
logej jx x 21 x 2
31 x 3
41 x 4
F (x )x 31
x 21
x1 1 x x 2 x 3
F'(x ) x 41
x 31
x 21
0 1 2x 3x 2
-
O x1x
F (x )
F (x1)
II- 5.7
F(x) =xn
x=0
x=x1
x= 0 x= x1
F (x )=x 2 x=0 x=x1
F (x )=ax n x=0 x =x1
n a
II-5.7 F (x )=x 2 x=0 x=x1
-
x
xO
O
F (x ) F1(x )
F2(x )
II- 5.8
x
-
II- 5.9
limx1` a1
x1
xdx
limx1d e1
x1dx exp` a x
-
II- 5.10
x1
p q
1
pq
xdx
=1
p
xdx
+1
q
xdx
-
F (x )
x
F (x )
O
(1) a=1
O 0.8
x
(2) a=0.8
F (x )
O
x
(3) a=1.2
1.2
II- 5.11
0
a
dx sin` aax a
II-5.11 F (x )= sin` aax
-
x
y
R
R
R
R O
II- 5.12
(x,y )= (0,0) R
R >0
x 2+y 2=R 2
y y= R 2 x 2
y x F (x )= R 2 x 2
R
II-5.12
-
-1
-
- 1.1
F (x ) n F (n)(x )
F (x ) x =x0
F (x )= F (x0)+F(1)(x0)` ax x0
+2!
F (2)(x0) ` ax x02 +
3!F (3)(x0) ` ax x0
3 +
= F (x0)+n =1
n !F (n)(x0) ` ax x0 n
-
- 1.2
loge` ax+1
F (x )= loge` ax+1 x =0
-
- 1.3
1x
1
F (x )=1x
1 x =0
-
- 1.4
y=F (x )= arctanx
1+t1
t=0
t=x 2 y =F (x )= arctanx
x=0
x=1
-
- 1.5
expx cosx sinx
exp x cosx sinx x=0
-
-2
-
- 2.1 II- 3.7
1 i
i2 =1 i = 1
17i 3
5i 2i
zRe zIm z
z = zRe +zIm i
zRe Rez
Rez = zRe
zIm Im z Im z = zIm
5+3i
i
zRe z Rez
zIm z Im z
-
- 2.2
II- 3.8
-2.2.1
-2.2.1
z0 z0r j jz0
z0r =j jz0 = z0Re2 + z0Im
2
z0 z0 arg z0
z0 = arg z0 = arctan` az0Rez0Im
z0
z0 = z0Re + z0Im i
= z0r` acosz0 + i sin z0
-
z1 z2
Rez1 = Rez2 Im z1 = Im z2
z1 z2
j jz1 =j jz2 argz1 = argz2
z1
z2
-
- 2.3
z F (z ) F (z )
FRe(z ) F (z ) ReF (z )
FIm(z ) F (z ) Im F (z )
-
Im zRez i
ReF (z )
- 2.4
F(z )=z 2+1 FRe(z )
F (z )= z 2+1 FRe(z ) z=zRe+ i
-2.4 F (z )= z 2+1
-
- 2.5
exp z =exp zRe` acoszIm +i sin zIm
z z=zRe + i zIm ` azRe = Rez zIm = Im z i = 1
exp z = exp zRe` acoszIm +i sin zIm exp z
-
- 2.6
z z1 z2 m
` aexp z1 ` aexp z2 = exp` az1+z2` aexp z1 ` aexp z2 = exp` az1 z2` aexp z m= exp` amz m
-
- 2.7
-
Im zRez
ReF (z )
z=i
- 2.8
-2.8 F (z )= exp z
-
Thank you for reading this document.
We can reach the end as "Web version."
My hope is to publish the "Whole virsion" some day..
-
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