a j 2 ā” + 1 ( j ā ā ) z = x + yj ( x , y ā ā ) z 1 + z 2 = ( x 1 + y 1 j ) + ( x 2 + y 2 j ) = ( x 1 + x 2 ) + ( y 1 + y 2 ) j z 1 z 2 = ( x 1 + y 1 j ) ( x 2 + y 2 j ) = x 1 x 2 + x 1 y 2 j + y 1 x 2 j + y 1 y 2 j 2 = ( x 1 x 2 + y 1 y 2 ) + ( x 1 y 2 + y 1 x 2 ) j z * = x - yi ( z 1 + z 2 ) * = z * 1 + z * 2 ( z 1 z 2 ) * = z * 1 z * 2 z 1
z 2 = z 1 z 2 *
z 2 z * 2 = ( x 1 x 2 - y 1 y 2 ) + ( y 1 x 2 - x 1 y 2 ) j
x 2 2 - y 2 2 ( x 2 ā y 2 ) | | z | | 2 ā” zz * = x 2 - y 2 | | z | | h ā” | zz * | = | | ( x + yj ) ( x - yj ) | | = | x 2 - y 2 | | | z | | 2 = Ā± | | z | | 2 h | | z 2 - z 1 | | h = | ( z 2 - z 1 ) ( z * 2 - z * 1 ) | ( z 2 - z 1 ) ( z * 2 - z * 1 ) = z 2 z * 2 - z 2 z * 1 - z 1 z 2 * + z 1 z * 1 = ( x 2 2 - y 2 2 ) - [ { ( x 1 x 2 - y 1 y 2 ) + ( x 1 y 2 - y 1 x 2 ) j } + { ( x 1 x 2 - y 1 y 2 ) + ( - x 1 y 2 + y 1 x 2 ) j } ] + ( x 2 1 - y 2 1 ) = ( x 2 2 - y 2 2 ) - 2 ( x 1 x 2 - y 1 y 2 ) + ( x 2 1 - y 2 1 ) = ( x 2 2 - 2x 1 x 2 + x 2 1 ) - ( y 2 2 - 2y 1 y 2 + y 2 1 ) = ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 | | z 2 - z 1 | | h = | ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 | | | z | | h = r | | z | | 2 h = r 2 | x 2 - y 2 | = r 2 l = rš | | w | | h = 1 | | w | | 2 h = 1 | x 2 - y 2 | = 1 l = š x 2 > y 2 ā¹ x 2 - y 2 = 1 y 2 > x 2 ā¹ y 2 - x 2 = 1 x 2 > y 2 , x > 0 ā¹ w 1 = cosh š + j sinh š cosh 2 š - sinh 2 š = 1 A = 1 + 0j B = cosh ( š¼ + š½ ) + j sinh ( š¼ + š½ ) A' = cosh ( - š¼ ) + j sinh ( - š¼ ) B' = cosh š½ + j sinh š½ | | B - A | | = | | B' - A' | | | | B - A | | 2 = | | B' - A' | | 2 ( cosh ( š¼ + š½ ) - 1 ) 2 - ( sinh ( š¼ + š½ ) - 0 ) 2 = ( cosh š½ - cosh š¼ ) 2 - ( sinh š½ + sinh š¼ ) 2 cosh 2 ( š¼ + š½ ) - 2 cosh ( š¼ + š½ ) + 1 - sinh 2 ( š¼ + š½ ) = cosh 2 š½ - 2 cosh š¼ cosh š½ + cosh 2 š¼ - sinh 2 š½ - 2 sinh š¼ sinh š½ - sinh 2 š¼ - 2 cosh ( š¼ + š½ ) + 2 = - 2 cosh š¼ cosh š½ - 2 sinh š¼ sinh š½ + 2 cosh ( š¼ + š½ ) = cosh š¼ cosh š½ + sinh š¼ sinh š½ cosh ( š¼ - š½ ) = cosh š¼ cosh š½ - sinh š¼ sinh š½ cosh 2š¼ = cosh 2 š¼ + sinh 2 š¼ = 2 sinh 2 š¼ + 1 = 2 cosh 2 š¼ - 1 sinh 2 š = cosh 2 š - 1 sinh 2 ( š¼ + š½ ) = cosh 2 ( š¼ + š½ ) - 1 = ( cosh š¼ cosh š½ + sinh š¼ sinh š½ ) 2 - 1 = cosh 2 š¼ cosh 2 š½ + 2 cosh š¼ cosh š½ sinh š¼ sinh š½ + sinh 2 š¼ sinh 2 š½ - cosh 2 š¼ + sinh 2 š¼ = cosh 2 š¼ ( cosh 2 š½ - 1 ) + sinh 2 š¼ ( sinh 2 š½ + 1 ) + 2 cosh š¼ cosh š½ sinh š¼ sinh š½ = cosh 2 š¼ sinh 2 š½ + 2 cosh š¼ sinh š½ sinh š¼ cosh š½ + sinh 2 š¼ cosh 2 š½ sinh 2 ( š¼ + š½ ) = ( cosh š¼ sinh š½ + sinh š¼ cosh š½ ) 2 sinh 2 ( 2š¼ ) = ( 2 sinh š¼ cosh š¼ ) 2 š¼ ā©¾ 0 ā¹ sinh ( 2š¼ ) ā©¾ 0 , sinh š¼ ā©¾ 0 , cosh š¼ ā©¾ 1 š¼ < 0 ā¹ sinh ( 2š¼ ) < 0 , sinh š¼ < 0 , cosh š¼ > 1 sinh ( 2š¼ ) = 2 sinh š¼ cosh š¼ sinh ( š¼ + š½ ) = cosh š¼ sinh š½ + sinh š¼ cosh š½ x = 2 ā¹ y = Ā± 1 cosh š¾ = 2 ( š¾ > 0 ) dl = | dx 2 - dy 2 | dl = | ( dx
dy ) 2 - 1 | dy x 2 - y 2 = 1 x 2 = y 2 + 1 x = y 2 + 1 dx
dy = y
y 2 + 1 š¾ 0 l = 1 ā« 0 | ( dx
dy ) 2 - 1 | dy = 1 ā« 0 | y 2
y 2 + 1 - 1 | dy = 1 ā« 0 | - 1
y 2 + 1 | dy = 1 ā« 0 1
y 2 + 1 dy y = tan t dy = 1
cos 2 t dt 1 ā« 0 ā š
4 ā« 0 š¾ 0 l = š
4 ā« 0 1
tan 2 t + 1 ā
1
cos 2 t dt = š
4 ā« 0 1
cos t dt = š
4 ā« 0 cos t
cos 2 t dt = š
4 ā« 0 cos t
1 - sin 2 t dt sin t = s ds = cos t dt š
4 ā« 0 ā 2
2 ā« 0 š¾ 0 l = 2
2 ā« 0 1
1 - s 2 ds = 2
2 ā« 0 - 1
( s - 1 ) ( s + 1 ) ds = - 1
2 2
2 ā« 0 ( 1
s - 1 - 1
s + 1 ) ds = - 1
2 [ ln | s - 1
s + 1 | ] 2
2 0 = - 1
2 ( ln | 2 -2
2
2 +2
2 | - ln | | - 1 | | ) = - 1
2 ( ln | 2 - 4 2 + 4
- 2 | - ln 1 ) = - 1
2 ln | 2 2 - 3 | š¾ 0 l = š¾ = - 1
2 ln ( 3 - 2 2 ) ā 0.88 y = tan t , sin t = s s = y cos t s = y
y 2 + 1 š 0 l = š = - 1
2 [ ln | s - 1
s + 1 | ] S 0 = - 1
2 ( ln | S - 1
S + 1 | - ln | | - 1 | | ) = - 1
2 ln | S - 1
S + 1 | s ( y ) ā” y
y 2 + 1 š ( y ) ā” - 1
2 ln | s ( y ) - 1
s ( y ) + 1 | š ( 0 ) = 0 š ( 1 ) = š¾ š' ( y ) ā©½ 1 š' ( 0 ) = 1 sinh š ( y ) = y x = y 2 + 1 š ( x ) = x 2 - 1
x š ( x ) = - 1
2 ln | š ( x ) - 1
š ( x ) + 1 | š ( 1 ) = 0 š ( 2 ) = š¾ š' ( 2 ) = 1 0 < X < 2 š' ( X ) > 1 (...) cosh š ( x ) = x 0 < š < š¾ ā¹ 0 < š < sinh š < 1 < cosh š < 2 š < sinh š 1 < sinh š
š sinh š
š < cosh š 1 < sinh š
š < cosh š lim š ā 0 1 ā©½ lim š ā 0 sinh š
š ā©½ lim š ā 0 cosh š 1 ā©½ lim š ā 0 sinh š
š ā©½ 1 lim š ā 0 sinh š
š = 1 lim š ā 0 cosh š - 1
š = lim š ā 0 cosh 2 š - 1
š ( cosh š + 1 ) = lim š ā 0 sinh 2 š
š ( cosh š + 1 ) = lim š ā 0 ( sinh š
š ā
sinh š
cosh š + 1 ) = 1 ā
( 0
2 ) = 0 d
dš ( sinh š ) = lim h ā 0 sinh ( š + h ) - sinh š
h = lim h ā 0 cosh š sinh h + sinh š cosh h - sinh š
h = lim h ā 0 cosh š sinh h + sinh š ( cosh h - 1 )
h = lim h ā 0 ( sinh h
h cosh š + cosh h - 1
h sinh š ) = 1 ā
cosh š + 0 ā
sinh š = cosh š d
dš ( cosh š ) = lim h ā 0 cosh ( š + h ) - cosh š
h = lim h ā 0 cosh š cosh h + sinh š sinh h - cosh š
h = lim h ā 0 cosh š ( cosh h - 1 ) + sinh š sinh h
h = lim h ā 0 ( cosh h - 1
h cosh š + sinh h
h sinh š ) = 0 ā
cosh š + 1 ā
sinh š = sinh š d
dx ( cosh x + sinh x ) = sinh x + cosh x cosh x + sinh x = e x cosh x - sinh x = e -x cosh x = e x + e -x
2 sinh x = e x - e -x
2 e x = ā ā n=0 x n
n! e -x = ā ā n=0 ( - x ) n
n! cosh x = ā ā n=0 x 2n
( 2n ) ! sinh x = ā ā n=0 x 2n+1
( 2n + 1 ) ! e jš = ā ā n=0 ( jš ) n
n! e jš = cosh š + j sinh š | | w | | h = 1 w 1 = e jš ( x 2 > y 2 , x > 0 ) w 3 = - e jš ( x 2 > y 2 , x < 0 ) w 2 = je jš ( y 2 > x 2 , y > 0 ) w 4 = - je jš ( y 2 > x 2 , y < 0 ) w = Ā± e jš or w = Ā± je jš | | w | | h = r w = Ā± re jš or w = Ā± rje jš