Теория функций комплексного переменного. - 27 стр.

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§4. äÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ ËÏÍÐÌÅËÓÎÙÈ ÆÕÎËÃÉÊ 27
ôÁÂÌÉÃÁ ÐÒÏÉÚ×ÏÄÎÙÈ
1.(z
n
)
0
= nz
n−1
. 2.(e
z
)
0
= e
z
.
3.(sin z)
0
= cos z. 4.(cos z)
0
= −sin z.
5.(tg z)
0
=
1
cos
2
z
. 6.(ctg z)
0
= −
1
sin
2
z
.
7.(sh z)
0
= ch z. 8.(ch z)
0
= sh z.
9.(th z)
0
=
1
ch
2
z
. 10.(cth z)
0
= −
1
sh
2
z
.
11.(Arcsin z)
0
=
1
√
1−z
2
. 12.(Arccos z)
0
= −
1
√
1−z
2
.
13.(Arctg z)
0
=
1
1+z
2
. 14.(Arcctg z)
0
= −
1
1+z
2
.
15.(Ln z)
0
=
1
z
.
úÁÍÅÔÉÍ, ÞÔÏ ÐÒÉ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÉ ÒÁÓÓÍÁÔÒÉ×ÁÀÔÓÑ ÔÏÌØËÏ ÏÄÎÏÚÎÁÞ-
ÎÙÅ ÆÕÎËÃÉÉ. ÷ ÒÁ×ÅÎÓÔ×ÁÈ (11)(15) ÐÏÎÉÍÁÅÍ ÌÅ×ÕÀ ÞÁÓÔØ ËÁË ÐÒÏÉÚ×ÏÄ-
ÎÕÀ ÏÔ ÐÒÏÉÚ×ÏÌØÎÏÊ ÏÄÎÏÚÎÁÞÎÏÊ ×ÅÔ×É ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÅÊ ÆÕÎËÃÉÉ, ×ÙÄÅ-
ÌÅÎÎÏÊ × ÏËÒÅÓÔÎÏÓÔÉ ÄÁÎÎÏÊ ÔÏÞËÉ.
ðÒÉÍÅÒ 1. îÁÊÔÉ ÇÄÅ ÄÉÆÆÅÒÅÎÃÉÒÕÅÍÁ ÆÕÎËÃÉÑ f (z) =
e
z
+ 1
e
z
− 1
É ÎÁÊÔÉ
ÅÅ ÐÒÏÉÚ×ÏÄÎÕÀ.
òÅÛÅÎÉÅ: æÕÎËÃÉÑ f (z) ÏÐÒÅÄÅÌÅÎÁ ×ÅÚÄÅ ËÒÏÍÅ ÔÏÞÅË, ÇÄÅ
e
z
− 1 = 0, e
z
= 1.
ðÏÌÏÖÉÍ
z = x + iy, e
z
= e
x+iy
= e
x
(cos y + i sin y).
ðÒÅÄÓÔÁ×ÉÍ ÞÉÓÌÏ 1 × ÔÒÉÇÏÎÏÍÅÔÒÉÞÅÓËÏÊ ÆÏÒÍÅ
1 = cos 2πk + i sin 2πk, k ∈ Z,
ÐÏÜÔÏÍÕ e
x
= 1, y = 2πk ÉÌÉ x = 0, y = 2πk, k ∈ Z. ïËÏÎÞÁÔÅÌØÎÏ ÐÏÌÕÞÉÍ
z = i2πk, k ∈ Z. ôÁËÉÍ ÏÂÒÁÚÏÍ, ÆÕÎËÃÉÑ f(z) ÏÐÒÅÄÅÌÅÎÁ ×ÓÀÄÕ ËÒÏÍÅ
ÔÏÞÅË z = i2πk, k ∈ Z.
ðÏ ÐÒÁ×ÉÌÕ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÑ ÞÁÓÔÎÏÇÏ Ä×ÕÈ ÆÕÎËÃÉÊ ÐÏÌÕÞÉÍ
f
0
(z) =
(e
z
+ 1)
0
(e
z
− 1) − (e
z
+ 1)(e
z
− 1)
0
(e
z
− 1)
2
=
=
e
z
(e
z
− 1) − e
z
(e
z
+ 1)
(e
z
− 1)
2
=
−2e
z
(e
z
− 1)
2
.
éÚ ÐÏÓÌÅÄÎÅÇÏ ÒÁ×ÅÎÓÔ×Á ×ÉÄÎÏ, ÞÔÏ ÆÕÎËÃÉÑ Ñ×ÌÑÅÔÓÑ ÄÉÆÆÅÒÅÎÃÉÒÕÅÍÏÊ
×Ï ×ÓÅÈ ÔÏÞËÁÈ, ÇÄÅ ÏÎÁ ÏÐÒÅÄÅÌÅÎÁ.
§4. äÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ ËÏÍÐÌÅËÓÎÙÈ ÆÕÎËÃÉÊ                                         27

  ôÁÂÌÉÃÁ ÐÒÏÉÚ×ÏÄÎÙÈ

               1.(z n )0 = nz n−1.             2.(ez )0 = ez .
               3.(sin z)0 = cos z.             4.(cos z)0 = − sin z.
               5.(tg z)0 = cos12 z .           6.(ctg z)0 = − sin12 z .
               7.(sh z)0 = ch z.               8.(ch z)0 = sh z.
               9.(th z)0 = ch12 z .            10.(cth z)0 = − sh12 z .
                                       1                                 1
               11.(Arcsin z)0 = √1−z      2.   12.(Arccos z)0 = − √1−z       2.
                                     1                                  1
               13.(Arctg z)0 = 1+z     2.      14.(Arcctg z)0 = − 1+z      2.
                          0    1
               15.(Ln z) = z .

úÁÍÅÔÉÍ, ÞÔÏ ÐÒÉ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÉ ÒÁÓÓÍÁÔÒÉ×ÁÀÔÓÑ ÔÏÌØËÏ ÏÄÎÏÚÎÁÞ-
ÎÙÅ ÆÕÎËÃÉÉ. ÷ ÒÁ×ÅÎÓÔ×ÁÈ (11) (15) ÐÏÎÉÍÁÅÍ ÌÅ×ÕÀ ÞÁÓÔØ ËÁË ÐÒÏÉÚ×ÏÄ-
ÎÕÀ ÏÔ ÐÒÏÉÚ×ÏÌØÎÏÊ ÏÄÎÏÚÎÁÞÎÏÊ ×ÅÔ×É ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÅÊ ÆÕÎËÃÉÉ, ×ÙÄÅ-
ÌÅÎÎÏÊ × ÏËÒÅÓÔÎÏÓÔÉ ÄÁÎÎÏÊ ÔÏÞËÉ.
                                                          ez + 1
    ðÒÉÍÅÒ 1. îÁÊÔÉ ÇÄÅ ÄÉÆÆÅÒÅÎÃÉÒÕÅÍÁ ÆÕÎËÃÉÑ f (z) = z        É ÎÁÊÔÉ
                                                          e −1
ÅÅ ÐÒÏÉÚ×ÏÄÎÕÀ.
    òÅÛÅÎÉÅ: æÕÎËÃÉÑ f (z) ÏÐÒÅÄÅÌÅÎÁ ×ÅÚÄÅ ËÒÏÍÅ ÔÏÞÅË, ÇÄÅ

                                  ez − 1 = 0, ez = 1.

ðÏÌÏÖÉÍ
                  z = x + iy, ez = ex+iy = ex (cos y + i sin y).
ðÒÅÄÓÔÁ×ÉÍ ÞÉÓÌÏ 1 × ÔÒÉÇÏÎÏÍÅÔÒÉÞÅÓËÏÊ ÆÏÒÍÅ

                          1 = cos 2πk + i sin 2πk, k ∈ Z,

ÐÏÜÔÏÍÕ ex = 1, y = 2πk ÉÌÉ x = 0, y = 2πk, k ∈ Z. ïËÏÎÞÁÔÅÌØÎÏ ÐÏÌÕÞÉÍ
z = i2πk, k ∈ Z. ôÁËÉÍ ÏÂÒÁÚÏÍ, ÆÕÎËÃÉÑ f (z) ÏÐÒÅÄÅÌÅÎÁ ×ÓÀÄÕ ËÒÏÍÅ
ÔÏÞÅË z = i2πk, k ∈ Z.
   ðÏ ÐÒÁ×ÉÌÕ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÑ ÞÁÓÔÎÏÇÏ Ä×ÕÈ ÆÕÎËÃÉÊ ÐÏÌÕÞÉÍ

  0      (ez + 1)0 (ez − 1) − (ez + 1)(ez − 1)0
 f (z) =                                        =
                        (ez − 1)2
                                            ez (ez − 1) − ez (ez + 1)    −2ez
                                          =                           = z       .
                                                    (ez − 1)2          (e − 1)2
éÚ ÐÏÓÌÅÄÎÅÇÏ ÒÁ×ÅÎÓÔ×Á ×ÉÄÎÏ, ÞÔÏ ÆÕÎËÃÉÑ Ñ×ÌÑÅÔÓÑ ÄÉÆÆÅÒÅÎÃÉÒÕÅÍÏÊ
×Ï ×ÓÅÈ ÔÏÞËÁÈ, ÇÄÅ ÏÎÁ ÏÐÒÅÄÅÌÅÎÁ.

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