sinh(i*z)=-i (уравнение)

Найду корень уравнения: sinh(i*z)=-i

Вы ввели

sinh(I*z) = -I

$$\sinh{\left(i z \right)} = - i$$

Подробное решение

Дано уравнение
$$\sinh{\left(i z \right)} = - i$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на i

Ур-ние превратится в
$$\sin{\left(z \right)} = -1$$
Это ур-ние преобразуется в
$$z = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$z = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Или
$$z = 2 \pi n - \frac{\pi}{2}$$
$$z = 2 \pi n + \frac{3 \pi}{2}$$
, где n - любое целое число

Быстрый ответ

     -pi 
z1 = ----
      2  

$$z_{1} = - \frac{\pi}{2}$$

     3*pi
z2 = ----
      2  

$$z_{2} = \frac{3 \pi}{2}$$

Сумма и произведение корней

сумма

  pi   3*pi
- -- + ----
  2     2  

$$- \frac{\pi}{2} + \frac{3 \pi}{2}$$

=

pi

$$\pi$$

произведение

-pi  3*pi
----*----
 2    2  

$$- \frac{\pi}{2} \frac{3 \pi}{2}$$

=

     2
-3*pi 
------
  4   

$$- \frac{3 \pi^{2}}{4}$$

Численный ответ

z1 = -14.1371667858125

z2 = 36.1283159497235

z3 = -76.9690203748894

z4 = -32.9867224188086

z5 = -70.6858343571487

z6 = -58.1194645939029

z7 = -76.9690195738024

z8 = 80.1106131368654

z9 = 73.8274268520838

z10 = -45.5530935911043

z11 = 48.6946859012172

z12 = -95.818575476176

z13 = 42.4115013353669

z14 = 86.3937984838325

z15 = 10.9955747360645

z16 = 92.6769843439965

z17 = 86.3937978869933

z18 = -14.1371674455661

z19 = 61.2610571125526

z20 = -7.85398149665124

z21 = 98.9601692809083

z22 = -89.535390750197

z23 = -58.1194639046052

z24 = 23.5619444059921

z25 = 29.8451297031011

z26 = 23.5619451518571

z27 = 54.9778710948428

z28 = 17.2787591562062

z29 = -1.57079639503667

z30 = -39.2699076683741

z31 = -20.4203527465087

z32 = -70.6858351534454

z33 = 48.6946870830469

z34 = 36.1283150875497

z35 = 92.6769830592094

z36 = 29.845130330036

z37 = 538.783139388541

z38 = -39.2699084145515

z39 = 23.5619437708833

z40 = -70.6858331259916

z41 = 98.9601690454399

z42 = -95.8185758680502

z43 = -64.4026498988255

z44 = 17.2787599560783

z45 = 61.2610563112167

z46 = -64.4026491641039

z47 = -1.57079643188553

z48 = 48.6946873020308

z49 = 73.8274274426229

z50 = 80.1106130902139

z51 = -14.1371668370864

z52 = 48.6946866365921

z53 = 67.54424230971

z54 = -89.5353901118113

z55 = 67.5442415586719

z56 = -83.2522042893833

z57 = -58.1194639976905

z58 = 4.71238874329685

z59 = -7.85398119154045

z60 = 36.1283157235346

z61 = -45.5530929624673

z62 = 54.9778718908148

z63 = -32.9867232184024

z64 = -39.2699069219675

z65 = -51.8362786893284

z66 = 67.5442408278864

z67 = 98.9601682515978

z68 = -51.8362791922783

z69 = -26.7035372004893

z70 = 80.1106122287081

z71 = 86.3937978309099

z72 = -1.57079581340397

z73 = -64.4026502975618

z74 = -20.4203520060805

z75 = -89.5353906059052

z76 = -20.420353265929

z77 = 73.8274274830848

z78 = 4.71239022926564

z79 = 4.7123894841958

z80 = 42.4115007162407

z81 = -102.101761026058

z82 = -26.7035379986821

z83 = -45.5530935025548

z84 = 92.6769837888103

z85 = 29.8451303231501

z86 = 42.4115007274741

z87 = -83.2522055723275

z88 = -83.2522048211133

z89 = -7.85398205280014

z90 = 10.9955739381756

z91 = -51.8362783335234

z92 = -95.8185763308148