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所以 Lnz和2Lnz的实部相同，但虚部不尽相同，故Lnz?2Lnz不正确。 2) Lnz?221Lnz 2解：设z?rei?

?i?????1???2?Lnz?Ln?re?lnr?i?2n??lnr?i?2n????? n?0,?1,?2,? ???2?2?2???

11111???Lnz?Ln?rei???lnr?i???2m???lnr?i??m?? m?0,?1,?2,? 22222?2?所以 Lnz和

1Lnz的实部相同，但虚部不尽相同，故Lnz2?2Lnz不正确。 2i18． 求e解：e1?i1?i?2，e11?i?4i，3和?1?i?的值。

?2?ee14?i?2?????e?cos?isin???ie

22?? e1?i?4?eeiLn3i?4???24??e?cos?isin??e?1?i?

44?2?141 3?ei?ei?ln3?i2n???e2n?eiln3?e2n??cos?ln3??isin?ln3??

?????iln2???2n???4? ?1?i??eiLn1?i?ei?e??????2n???4??cos?ln2??isin?ln2?? n?0,?1,?2,?

'19． 证明za???az'a?1，其中a为实数。

证明：如果a是整数，则z????e??e?aLnz?a'aLnz'aLnza'alnz'alnz?zaa1?aza?1 z1?aza?1 z 如果a不是整数，则z20． 证明：

1) chz?shz?1；

22????e??e?alnz?'?zaa11?1??1?证明：ch2z?sh2z???ez?e?z?????ez?e?z????e2z?e?2z?2???e2z?e?2z?2??1

44?2??2?2) shz?chz?ch2z；

222211?1??1?证明：shz?chz???ez?e?z?????ez?e?z????e2z?e?2z?2???e2z?e?2z?2?

44?2??2?2222?11?2e2z?2e?2z???e2z?e?2z??ch2z 423) sh?z1?z2??shz1chz2?chz1shz2，ch?z1?z2??chz1chz2?shz1shz2。 证明：shz1chz2?chz1shz2?

1z11z211z2e?e?z1e?e?z2?ez1?e?z1e?e?z2 222211?ez1ez2?e?z1ez2?ez1e?z2?e?z1e?z2?ez1ez2?e?z1ez2?ez1e?z2?e?z1e?z2 4411?ez1?z2?e?z1?z2?ez1?z2?e?z1?z2?ez1?z2?e?z1?z2?ez1?z2?e?z1?z2?sh?z1?z2? 4411z211z2chz1chz2?shz1shz2?ez1?e?z1e?e?z2?ez1?e?z1e?e?z2

222211?ez1ez2?e?z1ez2?ez1e?z2?e?z1e?z2?ez1ez2?e?z1ez2?ez1e?z2?e?z1e?z2 4411?ez1?z2?e?z1?z2?ez1?z2?e?z1?z2?ez1?z2?e?z1?z2?ez1?z2?e?z1?z2?ch?z1?z2? 44????????????????????????????????21． 解下列方程：

1) shz?0；

解：?shz?0 ?e?e2) chz?0；

解：?chz?0 ?e?e3) shz?i。

解：?shz?i ?e?ez?zz?zz?z?0 即e2z?0 ? z?in? n?0,?1,?2,?

??0 即e2z??1 ? z?i?n??1??? n?0,?1,?2,? 2??2i 即e2z?2iez?1?0 ? ez?i??2?0

z e?i ? z?i?2n??????? n?0,?1,?2,? 2?22． 证明?2.3.19?与?2.3.20?

?2.3.19? chiy?cosy，shiy?isiny

证明：chiy? ?1iy1?e?e?iy???cosy?isiny?cos??y??isin??y?? 221?cosy?isiny?cosy?isiny??cosy 21iy1?iy shiy??e?e???cosy?isiny?cos??y??isin??y??

221 ??cosy?isiny?cosy?isiny??isiny

2?ch?x?iy??chxcosy?ishxsiny?2.3.20? ?

??shx?iy?shxcosy?ichxsiny?证明：ch?x?iy??1x?iy1?e?e?x?iy???exeiy?e?xe?iy? 22 ?1x1e?cosy?isiny??e?x?cosy?isiny???ex?e?x?cosy?i?ex?e?x?siny 22???? ?chxcosy?ishxsiny

1x?iy1?e?e?x?iy???exeiy?e?xe?iy? 2211?ex?cosy?isiny??e?x?cosy?isiny???ex?e?x?cosy?i?ex?e?x?siny 22sh?x?iy???????shxcosy?ichxsiny

23． 证明：shz的反函数Acshz?Lnz?证明：设z?shw，则w?Arcshz。 z?shw??z2?1。

?1w?e?e?w? ? 2z?ew?e?w ? e2w?2zew?1?0 2 ? ew?z?z2?1 ? w?Lnz?z2?1 即Acshz?Lnz?z2?1

????24． 已知平面流速的复势f?z?为： 1)

?z?i?2；

32) z； 3)

1； 2z?1求流动的速度以及流线和等势线的方程。