z^8=1-i (уравнение)

Найду корень уравнения: z^8=1-i

Вы ввели

 8        
z  = 1 - I

$$z^{8} = 1 - i$$

Подробное решение

Дано уравнение
$$z^{8} = 1 - i$$
Т.к. степень в ур-нии равна = 8 и свободный член = 1 - i комплексное,
зн. действительных решений у соотв. ур-ния не существует

Остальные 8 корня(ей) являются комплексными.
сделаем замену:
$$w = z$$
тогда ур-ние будет таким:
$$w^{8} = 1 - i$$
Любое комплексное число можно представить так:
$$w = r e^{i p}$$
подставляем в уравнение
$$r^{8} e^{8 i p} = 1 - i$$
где
$$r = \sqrt[16]{2}$$
- модуль комплексного числа
Подставляем r:
$$e^{8 i p} = \frac{\sqrt{2} \left(1 - i\right)}{2}$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = \frac{\sqrt{2} \left(1 - i\right)}{2}$$
значит
$$\cos{\left(8 p \right)} = \frac{\sqrt{2}}{2}$$
и
$$\sin{\left(8 p \right)} = - \frac{\sqrt{2}}{2}$$
тогда
$$p = \frac{\pi N}{4} - \frac{\pi}{32}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для w
Значит, решением будет для w:
$$w_{1} = - \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
$$w_{2} = \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
$$w_{3} = - \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
$$w_{4} = \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
$$w_{5} = - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{6} = \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{7} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{8} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
делаем обратную замену
$$w = z$$
$$z = w$$

Тогда, окончательный ответ:
$$z_{1} = - \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
$$z_{2} = \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
$$z_{3} = - \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
$$z_{4} = \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
$$z_{5} = - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{6} = \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{7} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{8} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$

Быстрый ответ

       16___    /pi\     16___    /pi\
z1 = - \/ 2 *sin|--| - I*\/ 2 *cos|--|
                \32/              \32/

$$z_{1} = - \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$

     16___    /pi\     16___    /pi\
z2 = \/ 2 *sin|--| + I*\/ 2 *cos|--|
              \32/              \32/

$$z_{2} = \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$

       16___    /pi\     16___    /pi\
z3 = - \/ 2 *cos|--| + I*\/ 2 *sin|--|
                \32/              \32/

$$z_{3} = - \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$

     16___    /pi\     16___    /pi\
z4 = \/ 2 *cos|--| - I*\/ 2 *sin|--|
              \32/              \32/

$$z_{4} = \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$

       /   9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
       |  2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|
       |           \32/            \32/|            \32/            \32/
z5 = I*|- ------------- - -------------| + ------------- - -------------
       \        2               2      /         2               2      

$$z_{5} = - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)$$

       / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
       |2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|
       |         \32/            \32/|            \32/            \32/
z6 = I*|------------- - -------------| + ------------- + -------------
       \      2               2      /         2               2      

$$z_{6} = \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$

       / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
       |2    *sin|--|   2    *cos|--||   2    *cos|--|   2    *sin|--|
       |         \32/            \32/|            \32/            \32/
z7 = I*|------------- - -------------| - ------------- - -------------
       \      2               2      /         2               2      

$$z_{7} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)$$

       / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
       |2    *cos|--|   2    *sin|--||   2    *sin|--|   2    *cos|--|
       |         \32/            \32/|            \32/            \32/
z8 = I*|------------- + -------------| + ------------- - -------------
       \      2               2      /         2               2      

$$z_{8} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(\frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$

Численный ответ

z1 = 0.662480274400119 - 0.807234549989035*i

z2 = -0.662480274400119 + 0.807234549989035*i

z3 = 0.807234549989035 + 0.662480274400119*i

z4 = -0.807234549989035 - 0.662480274400119*i

z5 = -0.102356729874669 - 1.03924531873597*i

z6 = 1.03924531873597 - 0.102356729874669*i

z7 = 0.102356729874669 + 1.03924531873597*i

z8 = -1.03924531873597 + 0.102356729874669*i