sin(z)=-i (уравнение)

Найду корень уравнения: sin(z)=-i

Вы ввели

sin(z) = -I

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

Подробное решение

Дано уравнение
$$\sin{\left(z \right)} = - i$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$z = 2 \pi n + \operatorname{asin}{\left(- i \right)}$$
$$z = 2 \pi n + \pi - \operatorname{asin}{\left(- i \right)}$$
Или
$$z = 2 \pi n - i \log{\left(1 + \sqrt{2} \right)}$$
$$z = 2 \pi n + \pi + i \log{\left(1 + \sqrt{2} \right)}$$
, где n - любое целое число

Быстрый ответ

           /      ___\
z1 = -I*log\1 + \/ 2 /

$$z_{1} = - i \log{\left(1 + \sqrt{2} \right)}$$

               /      ___\
z2 = pi + I*log\1 + \/ 2 /

$$z_{2} = \pi + i \log{\left(1 + \sqrt{2} \right)}$$

Сумма и произведение корней

сумма

         /      ___\             /      ___\
0 - I*log\1 + \/ 2 / + pi + I*log\1 + \/ 2 /

$$\left(0 - i \log{\left(1 + \sqrt{2} \right)}\right) + \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$

=

pi

$$\pi$$

произведение

        /      ___\ /          /      ___\\
1*-I*log\1 + \/ 2 /*\pi + I*log\1 + \/ 2 //

$$1 \left(- i \log{\left(1 + \sqrt{2} \right)}\right) \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$

=

/           /      ___\\    /      ___\
\-pi*I + log\1 + \/ 2 //*log\1 + \/ 2 /

$$\left(\log{\left(1 + \sqrt{2} \right)} - i \pi\right) \log{\left(1 + \sqrt{2} \right)}$$

Численный ответ

z1 = 100.530964914873 - 0.881373587019543*i

z2 = -28.2743338823081 + 0.881373587019543*i

z3 = -100.530964914873 - 0.881373587019543*i

z4 = -94.2477796076938 - 0.881373587019543*i

z5 = 28.2743338823081 + 0.881373587019543*i

z6 = -9.42477796076938 + 0.881373587019543*i

z7 = 65.9734457253857 + 0.881373587019543*i

z8 = -84.8230016469244 + 0.881373587019543*i

z9 = -0.881373587019543*i

z10 = 97.3893722612836 + 0.881373587019543*i

z11 = 15.707963267949 + 0.881373587019543*i

z12 = -72.2566310325652 + 0.881373587019543*i

z13 = 21.9911485751286 + 0.881373587019543*i

z14 = 37.6991118430775 - 0.881373587019543*i

z15 = -3.14159265358979 + 0.881373587019543*i

z16 = -87.9645943005142 - 0.881373587019543*i

z17 = -65.9734457253857 + 0.881373587019543*i

z18 = 78.5398163397448 + 0.881373587019543*i

z19 = 47.1238898038469 + 0.881373587019543*i

z20 = -37.6991118430775 - 0.881373587019543*i

z21 = -31.4159265358979 - 0.881373587019543*i

z22 = -47.1238898038469 + 0.881373587019543*i

z23 = 6.28318530717959 - 0.881373587019543*i

z24 = -59.6902604182061 + 0.881373587019543*i

z25 = 91.106186954104 + 0.881373587019543*i

z26 = -21.9911485751286 + 0.881373587019543*i

z27 = -34.5575191894877 + 0.881373587019543*i

z28 = -12.5663706143592 - 0.881373587019543*i

z29 = -69.1150383789755 - 0.881373587019543*i

z30 = 40.8407044966673 + 0.881373587019543*i

z31 = 25.1327412287183 - 0.881373587019543*i

z32 = -53.4070751110265 + 0.881373587019543*i

z33 = -18.8495559215388 - 0.881373587019543*i

z34 = -6.28318530717959 - 0.881373587019543*i

z35 = -25.1327412287183 - 0.881373587019543*i

z36 = 59.6902604182061 + 0.881373587019543*i

z37 = -15.707963267949 + 0.881373587019543*i

z38 = -78.5398163397448 + 0.881373587019543*i

z39 = 56.5486677646163 - 0.881373587019543*i

z40 = -97.3893722612836 + 0.881373587019543*i

z41 = -91.106186954104 + 0.881373587019543*i

z42 = 94.2477796076938 - 0.881373587019543*i

z43 = 84.8230016469244 + 0.881373587019543*i

z44 = 81.6814089933346 - 0.881373587019543*i

z45 = -56.5486677646163 - 0.881373587019543*i

z46 = 50.2654824574367 - 0.881373587019543*i

z47 = -62.8318530717959 - 0.881373587019543*i

z48 = 53.4070751110265 + 0.881373587019543*i

z49 = 31.4159265358979 - 0.881373587019543*i

z50 = 34.5575191894877 + 0.881373587019543*i

z51 = 3.14159265358979 + 0.881373587019543*i

z52 = 75.398223686155 - 0.881373587019543*i

z53 = 72.2566310325652 + 0.881373587019543*i

z54 = 43.9822971502571 - 0.881373587019543*i

z55 = 18.8495559215388 - 0.881373587019543*i

z56 = -75.398223686155 - 0.881373587019543*i

z57 = 9.42477796076938 + 0.881373587019543*i

z58 = 62.8318530717959 - 0.881373587019543*i

z59 = 69.1150383789755 - 0.881373587019543*i

z60 = 87.9645943005142 - 0.881373587019543*i

z61 = -50.2654824574367 - 0.881373587019543*i

z62 = 12.5663706143592 - 0.881373587019543*i

z63 = -81.6814089933346 - 0.881373587019543*i

z64 = -40.8407044966673 + 0.881373587019543*i

z65 = -43.9822971502571 - 0.881373587019543*i