"Введение в матанализ", Комплексные числа
Задача 4(4)
Условие
Найти значение выражения: the fraction with numerator 13 plus 12 i and denominator 6 i minus 8 plus the fraction with numerator open paren 1 plus 2 i close paren squared and denominator 2 plus i
Решение
the fraction with numerator 13 plus 12 i and denominator 6 i minus 8 equals the fraction with numerator 13 plus 12 i and denominator negative 8 plus 6 i equals the fraction with numerator negative 104 plus 72 and denominator 100 plus the fraction with numerator negative 96 minus 78 and denominator 100 i equals negative 0.32 minus 1.74 i
the fraction with numerator 1 plus 4 i minus 4 and denominator 2 plus i equals the fraction with numerator negative 3 plus i and denominator 2 plus i equals negative 0.4 plus 2.2 i
Тогда, искомое число: negative 0.32 minus 1.74 i minus 0.4 plus 2.2 i equals negative 0.72 plus 0.46 i
Ответ: negative 0.72 plus 0.46 i
Задача 13(4)
Условие
Найти модуль числа z equals 1 plus the cosine of open paren 8 pi over 7 close paren plus i the sine of open paren 8 pi over 7 close paren
Решение
the absolute value of a plus b i equals the square root of a squared plus b squared
the absolute value of z squared equals 1 plus 2 the cosine of open paren 8 pi over 7 close paren plus the cosine squared of open paren 8 pi over 7 close paren plus the sine squared of open paren 8 pi over 7 close paren equals 2 plus 2 the cosine of open paren 8 pi over 7 close paren equals 4 times open paren the fraction with numerator 1 plus the cosine of open paren 4 pi over 7 close paren and denominator 2 close paren equals 4 the cosine squared of open paren 4 pi over 7 close paren
Ответ: the absolute value of z equals negative 2 the cosine of open paren 4 pi over 7 close paren
Задача 15(2)
Условие
Найти множество точек комплексной плоскости, заданное уравнением the absolute value of z minus i is less than the absolute value of z plus i
Модуль разности комплексных чисел - расстояние между точками, которые они обозначают. То есть, мы берем все точки, расстояние которых от i меньше, чем от negative i.
Тогда ответ: верхняя полуплоскость, не включая ось X, то есть: Y is greater than 0
Задача 18(6)
Условие
Найти аргумент комплексного числа z equals 1 plus the cosine of open paren pi over 7 close paren plus i the sine of open paren pi over 7 close paren
Решение
the absolute value of z squared equals open paren 1 plus the cosine of open paren pi over 7 close paren close paren squared plus the sine squared of open paren pi over 7 close paren equals 2 plus 2 the cosine of open paren pi over 7 close paren equals 4 the cosine squared of open paren pi over 14 close paren
the absolute value of z equals 2 the cosine of open paren pi over 7 close paren
z equals 2 the cosine of open paren pi over 14 close paren times open paren cosine phi plus i sine phi close paren
2 the cosine of open paren pi over 14 close paren cosine phi equals 1 plus the cosine of open paren pi over 7 close paren
the cosine of open paren pi over 14 minus phi close paren plus the cosine of open paren pi over 14 plus phi close paren equals 1 plus the cosine of open paren pi over 7 close paren
Ответ: arg z equals phi equals pi over 14 plus 2 pi n comma n is a member of the integers
Задача 30(3)
Условие
Записать z equals open paren the fraction with numerator i to the eighth power plus the square root of 3 i to the fifth power and denominator 4 close paren to the fifth powerв алгебраической форме.
Решение
z equals open paren the fraction with numerator 1 plus the square root of 3 i and denominator 4 close paren to the fifth power equals the fraction with numerator open paren one half plus i the fraction with numerator the square root of 3 and denominator 2 close paren to the fifth power and denominator 2 to the fifth power equals the fraction with numerator open paren the cosine of open paren pi over 3 close paren plus i the sine of open paren pi over 3 close paren close paren to the fifth power and denominator 32 equals the fraction with numerator the cosine of open paren 5 pi over 3 close paren and denominator 32 plus i the fraction with numerator the sine of open paren 5 pi over 3 close paren and denominator 32
Ответ: z equals 1 over 64 plus i the fraction with numerator the square root of 3 and denominator 64
Задача 31(2)
Условие
Записать z equals open paren the fraction with numerator the square root of 3 i plus 1 and denominator i minus 1 close paren to the sixth power в тригонометрической форме.
Решение
a equals the square root of 3 i plus 1 equals 2 times open paren the cosine of open paren pi over 3 close paren plus i the sine of open paren pi over 3 close paren close paren
b equals i minus 1 equals the square root of 2 times open paren the cosine of open paren 3 pi over 4 close paren plus i the sine of open paren 3 pi over 4 close paren close paren
z equals the fraction with numerator a to the sixth power and denominator b to the sixth power equals 2 to the sixth power times open paren cosine 2 pi plus i sine 2 pi close paren over the square root of 2 to the sixth power times open paren the cosine of open paren pi over 2 close paren plus i the sine of open paren pi over 2 close paren close paren equals 8 times open paren the cosine of open paren 3 pi over 2 close paren plus i the sine of open paren 3 pi over 2 close paren close paren
Задача 32(2, 7)
Условие
Найти все корни уравнений:
Решение, п. 2
z cubed equals 8 i
8 i equals 8 times open paren the cosine of open paren pi over 2 close paren plus i the sine of open paren pi over 2 close paren close paren
k sub i is a member of the set 0 comma 1 comma 2
По формуле корней комплексного числа:
r sub i equals the cube root of 8 times open paren the cosine of open paren the fraction with numerator pi over 2 plus 2 pi k sub i and denominator 3 close paren plus i the sine of open paren the fraction with numerator pi over 2 plus 2 pi k sub i and denominator 3 close paren close paren
Посчитаем углы:
the set alpha comma beta comma gamma equals the set pi over 6 comma 5 pi over 6 comma 9 pi over 2 equals 3 pi over 2
А потом и корни:
z sub 1 equals 2 times open paren cosine alpha plus i sine alpha close paren equals the square root of 3 plus i
z sub 2 equals 2 times open paren cosine beta plus i sine beta close paren equals the negative square root of 3 plus i
z sub 3 equals 2 times open paren cosine gamma plus i sine gamma close paren equals negative 2 i
Решение, п. 7
z to the sixth power plus 64 equals 0
z equals the square root of the cube root of 64
the cube root of 64 equals the set 4 times the fraction with numerator the square root of 3 plus i and denominator 2 comma 4 times the fraction with numerator the negative square root of 3 minus i and denominator 2 comma 4 times 1
z sub 1 comma 2 equals plus or minus 2
z sub 3 comma 4 equals 1 plus or minus the fraction with numerator i and denominator the square root of 3
z sub 5 comma 6 equals negative 1 plus or minus the fraction with numerator i and denominator the square root of 3
Теорема 2.
Утверджение
Изобразите на плоскости множество точек, заданное неравенством:
the real of open paren the fraction with numerator 2 minus i and denominator z minus the fraction with numerator 1 minus 2 i and denominator z bar close paren minus the imaginary of open paren the fraction with numerator 2 plus i and denominator z plus the fraction with numerator 1 plus 2 i and denominator z bar close paren is less than or equal to 2
Пусть z equals x plus i y
the fraction with numerator 2 minus i and denominator z minus the fraction with numerator 1 minus 2 i and denominator z bar equals the fraction with numerator 2 x minus y minus i times open paren x plus 2 y close paren and denominator x squared plus y squared minus the fraction with numerator x plus 2 y plus i times open paren y minus 2 x close paren and denominator x squared plus y squared equals
equals the fraction with numerator 2 x minus y minus x minus 2 y plus i times open paren negative x minus 2 y minus y plus 2 x close paren and denominator x squared plus y squared equals the fraction with numerator x minus 3 y plus i times open paren dot dot dot close paren and denominator x squared plus y squared
the fraction with numerator 2 plus i and denominator z plus the fraction with numerator 1 plus 2 i and denominator z bar equals the fraction with numerator 2 x plus y plus i times open paren x minus 2 y close paren and denominator x squared plus y squared plus the fraction with numerator x minus 2 y plus i times open paren 2 x plus y close paren and denominator x squared plus y squared equals
equals the fraction with numerator 3 x minus y plus i times open paren x minus 2 y plus 2 x plus y close paren and denominator x squared plus y squared equals the fraction with numerator open paren dot dot dot close paren plus i times open paren 3 x minus y close paren and denominator x squared plus y squared
the real of open paren the fraction with numerator 2 minus i and denominator z minus the fraction with numerator 1 minus 2 i and denominator z bar close paren equals the fraction with numerator x minus 3 y and denominator x squared plus y squared
the imaginary of open paren the fraction with numerator 2 plus i and denominator z plus the fraction with numerator 1 plus 2 i and denominator z bar close paren equals the fraction with numerator 3 x minus y and denominator x squared plus y squared
the fraction with numerator x minus 3 y minus 3 x plus y and denominator x squared plus y squared is less than or equal to 2
negative the fraction with numerator 2 times open paren x plus y close paren and denominator x squared plus y squared is less than or equal to 2
Это - вся плоскость, кроме окружности радиусом the fraction with numerator the square root of 2 and denominator 2с центром в open paren negative one half comma negative one half close paren: