sin(z)=pi*i (уравнение)

Найду корень уравнения: sin(z)=pi*i

Вы ввели

sin(z) = pi*I

$$\sin{\left(z \right)} = i \pi$$

Подробное решение

Дано уравнение
$$\sin{\left(z \right)} = i \pi$$
- это простейшее тригонометрическое ур-ние
Т.к. правая часть ур-ния
по модулю =

True

но sin
не может быть больше 1 или меньше -1
зн. решения у соотв. ур-ния не существует.

Быстрый ответ

z1 = I*asinh(pi)

$$z_{1} = i \operatorname{asinh}{\left(\pi \right)}$$

z2 = pi - I*asinh(pi)

$$z_{2} = \pi - i \operatorname{asinh}{\left(\pi \right)}$$

Сумма и произведение корней

сумма

0 + I*asinh(pi) + pi - I*asinh(pi)

$$\left(\pi - i \operatorname{asinh}{\left(\pi \right)}\right) + \left(0 + i \operatorname{asinh}{\left(\pi \right)}\right)$$

=

pi

$$\pi$$

произведение

1*I*asinh(pi)*(pi - I*asinh(pi))

$$1 i \operatorname{asinh}{\left(\pi \right)} \left(\pi - i \operatorname{asinh}{\left(\pi \right)}\right)$$

=

(pi*I + asinh(pi))*asinh(pi)

$$\left(\operatorname{asinh}{\left(\pi \right)} + i \pi\right) \operatorname{asinh}{\left(\pi \right)}$$

Численный ответ

z1 = -72.2566310325652 - 1.86229574331085*i

z2 = -53.4070751110265 - 1.86229574331085*i

z3 = 43.9822971502571 + 1.86229574331085*i

z4 = 100.530964914873 + 1.86229574331085*i

z5 = -97.3893722612836 - 1.86229574331085*i

z6 = 37.6991118430775 + 1.86229574331085*i

z7 = 50.2654824574367 + 1.86229574331085*i

z8 = 31.4159265358979 + 1.86229574331085*i

z9 = 53.4070751110265 - 1.86229574331085*i

z10 = -78.5398163397448 - 1.86229574331085*i

z11 = -6.28318530717959 + 1.86229574331085*i

z12 = -15.707963267949 - 1.86229574331085*i

z13 = -65.9734457253857 - 1.86229574331085*i

z14 = -12.5663706143592 + 1.86229574331085*i

z15 = -21.9911485751286 - 1.86229574331085*i

z16 = 91.106186954104 - 1.86229574331085*i

z17 = -34.5575191894877 - 1.86229574331085*i

z18 = 21.9911485751286 - 1.86229574331085*i

z19 = 6.28318530717959 + 1.86229574331085*i

z20 = -56.5486677646163 + 1.86229574331085*i

z21 = 65.9734457253857 - 1.86229574331085*i

z22 = -94.2477796076938 + 1.86229574331085*i

z23 = -37.6991118430775 + 1.86229574331085*i

z24 = 69.1150383789755 + 1.86229574331085*i

z25 = -43.9822971502571 + 1.86229574331085*i

z26 = -69.1150383789755 + 1.86229574331085*i

z27 = 56.5486677646163 + 1.86229574331085*i

z28 = -3.14159265358979 - 1.86229574331085*i

z29 = 34.5575191894877 - 1.86229574331085*i

z30 = 12.5663706143592 + 1.86229574331085*i

z31 = 59.6902604182061 - 1.86229574331085*i

z32 = -2.60930772597983e-40 + 1.86229574331085*i

z33 = 87.9645943005142 + 1.86229574331085*i

z34 = -50.2654824574367 + 1.86229574331085*i

z35 = 15.707963267949 - 1.86229574331085*i

z36 = 81.6814089933346 + 1.86229574331085*i

z37 = 84.8230016469244 - 1.86229574331085*i

z38 = 62.8318530717959 + 1.86229574331085*i

z39 = 25.1327412287183 + 1.86229574331085*i

z40 = -25.1327412287183 + 1.86229574331085*i

z41 = 28.2743338823081 - 1.86229574331085*i

z42 = 9.42477796076938 - 1.86229574331085*i

z43 = -28.2743338823081 - 1.86229574331085*i

z44 = 78.5398163397448 - 1.86229574331085*i

z45 = -91.106186954104 - 1.86229574331085*i

z46 = 18.8495559215388 + 1.86229574331085*i

z47 = 75.398223686155 + 1.86229574331085*i

z48 = -59.6902604182061 - 1.86229574331085*i

z49 = -31.4159265358979 + 1.86229574331085*i

z50 = -100.530964914873 + 1.86229574331085*i

z51 = -47.1238898038469 - 1.86229574331085*i

z52 = 97.3893722612836 - 1.86229574331085*i

z53 = -75.398223686155 + 1.86229574331085*i

z54 = 40.8407044966673 - 1.86229574331085*i

z55 = 94.2477796076938 + 1.86229574331085*i

z56 = -87.9645943005142 + 1.86229574331085*i

z57 = 4.9290750719775e-42 + 1.86229574331085*i

z58 = 72.2566310325652 - 1.86229574331085*i

z59 = -9.42477796076938 - 1.86229574331085*i

z60 = -81.6814089933346 + 1.86229574331085*i