上帝公式——Euler's formula

def get_value(n):
    x = (1 + 1j) ** n
    return x.real, x.imag


def plot_arrow(x, y, ax=plt, lines=True):
    if lines:
        ax.vlines(x, 0, y, 'r', '--')
        ax.hlines(y, 0, x, 'r', '--')
    ax.arrow(0, 0, x, y,
             color='cyan',
             head_width=0.2,
             fc='y',
             length_includes_head=True
             )


def plot_text(x, y, ax=plt):
    valign, halign = None, None
    if y > 0:
        text_y = f'{y}i'
        if x != 0: text_y = '+' + text_y
        y += .2
        valign = 'bottom'
    elif y == 0:
        text_y = ''
        y += .2
        valign = 'bottom'
    else:
        text_y = f'{y}i'
        y -= .2
        valign = 'top'

    if x > 0:
        text_x = f'{x}'
        x += .2
        halign = 'left'
    elif x == 0:
        text_x = ''
        x += .2
        halign = 'left'
    else:
        text_x = f'{x}'
        x -= .2
        halign = 'right'

    ax.text(x=x, y=y,  #文本x、y轴坐标
            s=f'${text_x}{text_y}$',  #文本内容
            fontdict=dict(fontsize=12),  #字体属性字典
            verticalalignment=valign,
            horizontalalignment=halign,
            bbox={  #添加文字背景色
                'facecolor': 'white',  #填充色
                'edgecolor': 'gray',  #外框色
                'alpha': 0.3,  #框透明度
                'pad': .1,  #本文与框周围距离
                'boxstyle': 'round'
            }
            )


# 绘图全局设置
plt.rcParams.update({
    # 'font.size': 15,
    'font.family': ['Times New Roman', 'SimSun']
})

limit_coor_num = 200

plt.hlines(0, -limit_coor_num, limit_coor_num, '#ccc', '--')
plt.vlines(0, -limit_coor_num, limit_coor_num, '#ccc', '--')

plt.xlim([-limit_coor_num, limit_coor_num])
plt.ylim([-limit_coor_num, limit_coor_num])
plt.yticks(
    [i for i in range(-limit_coor_num, limit_coor_num + 1) if i % 40 == 0],
    [f'{i}i' for i in range(-limit_coor_num, limit_coor_num + 1) if i % 40 == 0]
)

plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.title('Complex Plane')
# plt.grid()

# point_list = [(1, 0), (0, 1), (-1, 0), (0, -1)]
# point_list = [(-1, 2), (2, -1), (1, 1)]

# for x, y in point_list:
#     plot_arrow(x, y)
#     plot_text(x, y)

# for i in range(8):
#     x, y = get_value(i)
#     plot_arrow(x, y, lines=False)
#     plot_text(x, y)

points_plt = [get_value(i) for i in np.linspace(0, 100, 1000)]
x, y = zip(*points_plt)
plt.scatter(x, y)

# plot_arrow(0, -2, lines=False)
# plot_text(0, -2)

plt.show()

def get_value(n, p):
    x = np.power((1 + 3.14j / n), p)
    return x.real, x.imag

plt.xlim([-3, 3])
plt.ylim([0, 4])
plt.yticks(
    [i for i in range(0, 5) if i % 2 == 0],
    [f'{i}i' for i in range(0, 5) if i % 2 == 0]
)

for i in [1, 2, 5, 10, 20, 10000]:
    point_coor = [get_value(i, j) for j in range(1, i + 1)]
    x, y = zip(*point_coor)
    plt.scatter(x, y, s=(1 / i) * 100, marker='*')
    plt.plot(x, y, label=f'n={i}')

# x、y轴等比例
ax = plt.gca()
ax.set_aspect(1)

plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.title(r'$e^{i\pi} = \underset{n\rightarrow \infty}{\lim}\left( 1+\frac{i\pi}{n} \right) ^n=-1$')
plt.legend(ncol=3, loc='lower center', bbox_to_anchor=(0.5, -0.4))
plt.show()

def get_value(n, p, theta):
    x = np.power((1 + theta * 1j / n), p)
    return x.real, x.imag

fig, ax_arr = plt.subplots(2, 2)
fig.subplots_adjust(hspace=0.5)

theta_list = [1 / 2, 2 / 3, 5 / 4, 7 / 4]
sub_title = [r'\frac{1}{2}', r'\frac{2}{3}', r'\frac{5}{4}', r'\frac{7}{4}']
for i in range(ax_arr.shape[0]):
    for j in range(ax_arr.shape[1]):
        cur_ax = ax_arr[i, j]

        for k in [1, 2, 5, 10, 20, 10000]:
            point_coor = [get_value(k, v, theta_list[i * 2 + j] * np.pi) for v in range(1, k + 1)]
            x, y = zip(*point_coor)
            cur_ax.scatter(x, y, s=(1 / k) * 100, marker='*')
            cur_ax.plot(x, y, label=f'n={i}')

        cur_ax.axvline(0, color='#ccc', linestyle='--')
        cur_ax.axhline(0, color='#ccc', linestyle='--')

        # x、y轴等比例
        ax = plt.gca()
        ax.set_aspect(1)

        if j == 0:
            cur_ax.set_ylabel('Imaginary')
        if i == 1:
            cur_ax.set_xlabel('Real')

        cur_ax.set_title(f'$' + sub_title[i * 2 + j] + '\pi$')


lines, labels = fig.axes[-1].get_legend_handles_labels()  # 获取最后一个子图的图例
fig.legend(lines, labels, ncol=3, loc='upper center', bbox_to_anchor=(0.5, 0.03))
plt.tight_layout()
plt.show()