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..

IPAexIPA

IPAexIPAIPA

IPAV1.0

20158URL

http://ipafont.ipa.go.jp/ipafont/download.html

IPAv1.0

1 1

1

1.

2.

3.

4.

5. PDF

6.

7.

8.

1.

2.

3.

4.

5.

6. 32

()

7.

31

1. 47

(1)

(a)

;

(b)

(2)

(3)

(4)

(5)

2.6

(1)

(2)

(3)

3.

4.

1.

2.

1. IPA

2.