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z^8=1+i (уравнение)

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    Найду корень уравнения: z^8=1+i

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    комплексные числа z^8=1+i

    Решение

    Вы ввели [src]
     8        
z  = 1 + I
    $$z^{8} = 1 + i$$
    Упростить
    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:
    $$e^{8 i p} = \frac{\sqrt{2} \cdot \left(1 + i\right)}{2}$$
    Используя формулу Эйлера, найдём корни для p
    $$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = \frac{\sqrt{2} \cdot \left(1 + i\right)}{2}$$
    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:
    $$w_{1} = - \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
    $$w_{2} = \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
    $$w_{3} = - \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
    $$w_{4} = \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
    $$w_{5} = - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
    $$w_{6} = \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
    $$w_{7} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
    $$w_{8} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
    делаем обратную замену
    $$w = z$$
    $$z = w$$

    Тогда, окончательный ответ:
    $$z_{1} = - \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
    $$z_{2} = \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
    $$z_{3} = - \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
    $$z_{4} = \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
    $$z_{5} = - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
    $$z_{6} = \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
    $$z_{7} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
    $$z_{8} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
    График
    Быстрый ответ [src]
           16___    /pi\     16___    /pi\
z1 = - \/ 2 *sin|--| + I*\/ 2 *cos|--|
                \32/              \32/
    $$z_{1} = - \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
    Упростить
         16___    /pi\     16___    /pi\
z2 = \/ 2 *sin|--| - I*\/ 2 *cos|--|
              \32/              \32/
    $$z_{2} = \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}$$
    Упростить
           16___    /pi\     16___    /pi\
z3 = - \/ 2 *cos|--| - I*\/ 2 *sin|--|
                \32/              \32/
    $$z_{3} = - \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
    Упростить
         16___    /pi\     16___    /pi\
z4 = \/ 2 *cos|--| + I*\/ 2 *sin|--|
              \32/              \32/
    $$z_{4} = \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}$$
    Упростить
           / 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
    $$z_{5} = - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(\frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$
    Упростить
           / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
       |2    *sin|--|   2    *cos|--||   2    *cos|--|   2    *sin|--|
       |         \32/            \32/|            \32/            \32/
z6 = I*|------------- - -------------| + ------------- + -------------
       \      2               2      /         2               2
    $$z_{6} = \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)$$
    Упростить
           / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
       |2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|
       |         \32/            \32/|            \32/            \32/
z7 = I*|------------- - -------------| - ------------- - -------------
       \      2               2      /         2               2
    $$z_{7} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$
    Упростить
           /   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
    $$z_{8} = - \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)$$
    Упростить
    Сумма и произведение корней [src]
    сумма
                                                                                                                                                      / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\     / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\     / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\     /   9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\
                                                                                                                                                  |2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|     |2    *sin|--|   2    *cos|--||   2    *cos|--|   2    *sin|--|     |2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|     |  2    *cos|--|   2    *sin|--||   2    *sin|--|   2    *cos|--|
      16___    /pi\     16___    /pi\   16___    /pi\     16___    /pi\     16___    /pi\     16___    /pi\   16___    /pi\     16___    /pi\     |         \32/            \32/|            \32/            \32/     |         \32/            \32/|            \32/            \32/     |         \32/            \32/|            \32/            \32/     |           \32/            \32/|            \32/            \32/
0 + - \/ 2 *sin|--| + I*\/ 2 *cos|--| + \/ 2 *sin|--| - I*\/ 2 *cos|--| + - \/ 2 *cos|--| - I*\/ 2 *sin|--| + \/ 2 *cos|--| + I*\/ 2 *sin|--| + I*|------------- + -------------| + ------------- - ------------- + I*|------------- - -------------| + ------------- + ------------- + I*|------------- - -------------| - ------------- - ------------- + I*|- ------------- - -------------| + ------------- - -------------
               \32/              \32/            \32/              \32/              \32/              \32/            \32/              \32/     \      2               2      /         2               2           \      2               2      /         2               2           \      2               2      /         2               2           \        2               2      /         2               2
    $$\left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right) - \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} - i \left(\frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right) - i \left(- \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right) - i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right)$$
    =
      / 9/16    /pi\    9/16    /pi\\     / 9/16    /pi\    9/16    /pi\\     / 9/16    /pi\    9/16    /pi\\     /   9/16    /pi\    9/16    /pi\\
  |2    *cos|--|   2    *sin|--||     |2    *cos|--|   2    *sin|--||     |2    *sin|--|   2    *cos|--||     |  2    *cos|--|   2    *sin|--||
  |         \32/            \32/|     |         \32/            \32/|     |         \32/            \32/|     |           \32/            \32/|
I*|------------- + -------------| + I*|------------- - -------------| + I*|------------- - -------------| + I*|- ------------- - -------------|
  \      2               2      /     \      2               2      /     \      2               2      /     \        2               2      /
    $$i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right) + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right) + i \left(- \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right) + i \left(\frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$
    произведение
                                                                                                                                                  /  / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\\ /  / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\\ /  / 9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\\ /  /   9/16    /pi\    9/16    /pi\\    9/16    /pi\    9/16    /pi\\
                                                                                                                                              |  |2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|| |  |2    *sin|--|   2    *cos|--||   2    *cos|--|   2    *sin|--|| |  |2    *cos|--|   2    *sin|--||   2    *cos|--|   2    *sin|--|| |  |  2    *cos|--|   2    *sin|--||   2    *sin|--|   2    *cos|--||
  /  16___    /pi\     16___    /pi\\ /16___    /pi\     16___    /pi\\ /  16___    /pi\     16___    /pi\\ /16___    /pi\     16___    /pi\\ |  |         \32/            \32/|            \32/            \32/| |  |         \32/            \32/|            \32/            \32/| |  |         \32/            \32/|            \32/            \32/| |  |           \32/            \32/|            \32/            \32/|
1*|- \/ 2 *sin|--| + I*\/ 2 *cos|--||*|\/ 2 *sin|--| - I*\/ 2 *cos|--||*|- \/ 2 *cos|--| - I*\/ 2 *sin|--||*|\/ 2 *cos|--| + I*\/ 2 *sin|--||*|I*|------------- + -------------| + ------------- - -------------|*|I*|------------- - -------------| + ------------- + -------------|*|I*|------------- - -------------| - ------------- - -------------|*|I*|- ------------- - -------------| + ------------- - -------------|
  \           \32/              \32// \         \32/              \32// \           \32/              \32// \         \32/              \32// \  \      2               2      /         2               2      / \  \      2               2      /         2               2      / \  \      2               2      /         2               2      / \  \        2               2      /         2               2      /
    $$1 \left(- \sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}\right) \left(\sqrt[16]{2} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \cos{\left(\frac{\pi}{32} \right)}\right) \left(- \sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}\right) \left(\sqrt[16]{2} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} i \sin{\left(\frac{\pi}{32} \right)}\right) \left(- \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(\frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)\right) \left(\frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right) \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)\right) \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right)$$
    =
           -3*pi*I
       -------
  ___     4   
\/ 2 *e
    $$\sqrt{2} e^{- \frac{3 i \pi}{4}}$$
    Численный ответ [src]
    z1 = 1.03924531873597 + 0.102356729874669*i
    z2 = -0.807234549989035 + 0.662480274400119*i
    z3 = 0.102356729874669 - 1.03924531873597*i
    z4 = -0.102356729874669 + 1.03924531873597*i
    z5 = -1.03924531873597 - 0.102356729874669*i
    z6 = 0.807234549989035 - 0.662480274400119*i
    z7 = 0.662480274400119 + 0.807234549989035*i
    z8 = -0.662480274400119 - 0.807234549989035*i