第2章: 正則関数と複素微積分

import numpy as np import matplotlib.pyplot as plt from scipy.misc import derivative def f_analytic(z): """正則関数: f(z) = z^2""" return z**2 def f_not_analytic(z): """非正則関数: f(z) = z̄ (複素共役)""" return np.conj(z) # 実部と虚部を分離 def extract_uv(f, x, y): z = x + 1j*y w = f(z) return w.real, w.imag # 偏微分の数値計算 def check_cauchy_riemann(f, x0, y0, h=1e-5): u, v = extract_uv(f, x0, y0) # ∂u/∂x u_xp, _ = extract_uv(f, x0+h, y0) u_xm, _ = extract_uv(f, x0-h, y0) du_dx = (u_xp - u_xm) / (2*h) # ∂u/∂y u_yp, _ = extract_uv(f, x0, y0+h) u_ym, _ = extract_uv(f, x0, y0-h) du_dy = (u_yp - u_ym) / (2*h) # ∂v/∂x _, v_xp = extract_uv(f, x0+h, y0) _, v_xm = extract_uv(f, x0-h, y0) dv_dx = (v_xp - v_xm) / (2*h) # ∂v/∂y _, v_yp = extract_uv(f, x0, y0+h) _, v_ym = extract_uv(f, x0, y0-h) dv_dy = (v_yp - v_ym) / (2*h) return du_dx, du_dy, dv_dx, dv_dy # テスト点 x0, y0 = 1.5, 2.0 print("=== 正則関数: f(z) = z^2 ===") du_dx, du_dy, dv_dx, dv_dy = check_cauchy_riemann(f_analytic, x0, y0) print(f"∂u/∂x = {du_dx:.6f}") print(f"∂v/∂y = {dv_dy:.6f}") print(f"∂u/∂x - ∂v/∂y = {du_dx - dv_dy:.6e} (should be ~0)") print(f"\n∂u/∂y = {du_dy:.6f}") print(f"-∂v/∂x = {-dv_dx:.6f}") print(f"∂u/∂y - (-∂v/∂x) = {du_dy - (-dv_dx):.6e} (should be ~0)") print("\n\n=== 非正則関数: f(z) = z̄ ===") du_dx, du_dy, dv_dx, dv_dy = check_cauchy_riemann(f_not_analytic, x0, y0) print(f"∂u/∂x = {du_dx:.6f}") print(f"∂v/∂y = {dv_dy:.6f}") print(f"∂u/∂x - ∂v/∂y = {du_dx - dv_dy:.6f} (NOT ~0)") print(f"\n∂u/∂y = {du_dy:.6f}") print(f"-∂v/∂x = {-dv_dx:.6f}") print(f"∂u/∂y - (-∂v/∂x) = {du_dy - (-dv_dx):.6f} (NOT ~0)") # 可視化 x = np.linspace(-2, 2, 100) y = np.linspace(-2, 2, 100) X, Y = np.meshgrid(x, y) Z = X + 1j*Y # 正則関数 W_analytic = f_analytic(Z) U_analytic = W_analytic.real V_analytic = W_analytic.imag # 非正則関数 W_not = f_not_analytic(Z) U_not = W_not.real V_not = W_not.imag fig, axes = plt.subplots(2, 2, figsize=(14, 12)) # 正則関数: 実部 im = axes[0, 0].contourf(X, Y, U_analytic, levels=20, cmap='RdBu_r') axes[0, 0].contour(X, Y, V_analytic, levels=10, colors='green', linewidths=1, alpha=0.5) axes[0, 0].plot(x0, y0, 'ko', markersize=10) axes[0, 0].set_xlabel('x', fontsize=12) axes[0, 0].set_ylabel('y', fontsize=12) axes[0, 0].set_title('正則関数 $f(z)=z^2$: 実部(色), 虚部(緑線)', fontsize=12) plt.colorbar(im, ax=axes[0, 0]) # 正則関数: 虚部 im = axes[0, 1].contourf(X, Y, V_analytic, levels=20, cmap='RdBu_r') axes[0, 1].contour(X, Y, U_analytic, levels=10, colors='red', linewidths=1, alpha=0.5) axes[0, 1].plot(x0, y0, 'ko', markersize=10) axes[0, 1].set_xlabel('x', fontsize=12) axes[0, 1].set_ylabel('y', fontsize=12) axes[0, 1].set_title('正則関数 $f(z)=z^2$: 虚部(色), 実部(赤線)', fontsize=12) plt.colorbar(im, ax=axes[0, 1]) # 非正則関数: 実部 im = axes[1, 0].contourf(X, Y, U_not, levels=20, cmap='RdBu_r') axes[1, 0].contour(X, Y, V_not, levels=10, colors='green', linewidths=1, alpha=0.5) axes[1, 0].plot(x0, y0, 'ko', markersize=10) axes[1, 0].set_xlabel('x', fontsize=12) axes[1, 0].set_ylabel('y', fontsize=12) axes[1, 0].set_title('非正則関数 $f(z)=\\bar{z}$: 実部(色), 虚部(緑線)', fontsize=12) plt.colorbar(im, ax=axes[1, 0]) # 非正則関数: 虚部 im = axes[1, 1].contourf(X, Y, V_not, levels=20, cmap='RdBu_r') axes[1, 1].contour(X, Y, U_not, levels=10, colors='red', linewidths=1, alpha=0.5) axes[1, 1].plot(x0, y0, 'ko', markersize=10) axes[1, 1].set_xlabel('x', fontsize=12) axes[1, 1].set_ylabel('y', fontsize=12) axes[1, 1].set_title('非正則関数 $f(z)=\\bar{z}$: 虚部(色), 実部(赤線)', fontsize=12) plt.colorbar(im, ax=axes[1, 1]) plt.suptitle('正則関数と非正則関数の等高線 (直交性の検証)', fontsize=14) plt.tight_layout() plt.show()

import numpy as np import matplotlib.pyplot as plt # 複素微分の数値計算 def complex_derivative(f, z, h=1e-7): return (f(z + h) - f(z - h)) / (2*h) # テスト関数 functions = { 'z^2': lambda z: z**2, 'z^3': lambda z: z**3, 'e^z': lambda z: np.exp(z), 'sin(z)': lambda z: np.sin(z), '1/z': lambda z: 1/z, } # 理論的な導関数 derivatives_theory = { 'z^2': lambda z: 2*z, 'z^3': lambda z: 3*z**2, 'e^z': lambda z: np.exp(z), 'sin(z)': lambda z: np.cos(z), '1/z': lambda z: -1/z**2, } # テスト点 z0 = 1.5 + 2.0j print("複素微分の計算と検証:") print("=" * 60) for name, f in functions.items(): f_prime_numerical = complex_derivative(f, z0) f_prime_theoretical = derivatives_theory[name](z0) error = np.abs(f_prime_numerical - f_prime_theoretical) print(f"\nf(z) = {name}") print(f" z = {z0}") print(f" 数値微分: f'(z) = {f_prime_numerical:.6f}") print(f" 理論値: f'(z) = {f_prime_theoretical:.6f}") print(f" 誤差: {error:.2e}") # 可視化: ベクトル場としての導関数 x = np.linspace(-2, 2, 15) y = np.linspace(-2, 2, 15) X, Y = np.meshgrid(x, y) Z = X + 1j*Y fig, axes = plt.subplots(2, 3, figsize=(16, 10)) axes = axes.flatten() for idx, (name, f) in enumerate(functions.items()): if idx >= len(axes): break ax = axes[idx] # 関数値 W = f(Z) # 導関数(ベクトル場) F_prime = np.array([[complex_derivative(f, z) for z in row] for row in Z]) # ベクトル場の可視化 U = F_prime.real V = F_prime.imag magnitude = np.sqrt(U**2 + V**2) # 色で大きさを表示 im = ax.contourf(X, Y, magnitude, levels=20, cmap='viridis', alpha=0.6) # ベクトル場 skip = 2 ax.quiver(X[::skip, ::skip], Y[::skip, ::skip], U[::skip, ::skip], V[::skip, ::skip], magnitude[::skip, ::skip], cmap='hot', scale=50, width=0.003) ax.set_xlabel('Re(z)', fontsize=10) ax.set_ylabel('Im(z)', fontsize=10) ax.set_title(f"$f'(z)$ for $f(z) = {name}$", fontsize=12) ax.axis('equal') ax.grid(True, alpha=0.3) plt.colorbar(im, ax=ax, label="$|f'(z)|$") # 最後のサブプロットを削除(5つしか関数がないので) fig.delaxes(axes[5]) plt.suptitle('正則関数の導関数(ベクトル場表示)', fontsize=14) plt.tight_layout() plt.show()

import numpy as np import matplotlib.pyplot as plt from scipy.integrate import quad def complex_line_integral(f, z_t, t_range, n_points=1000): """ 複素線積分の数値計算 f: 被積分関数 z_t: 経路のパラメータ表示 z(t) t_range: (t_start, t_end) """ t = np.linspace(t_range[0], t_range[1], n_points) dt = t[1] - t[0] # 経路上の点 z = z_t(t) # 導関数 dz/dt (数値微分) dz_dt = np.gradient(z, dt) # 被積分関数の値 f_z = f(z) # 積分 (台形則) integral = np.trapz(f_z * dz_dt, dx=dt) return integral, z # テスト関数 f1 = lambda z: z f2 = lambda z: z**2 f3 = lambda z: 1/z # 経路1: 原点から1+iへの直線 path1 = lambda t: t * (1 + 1j) t_range1 = (0, 1) print("=== 経路1: 直線 (0 → 1+i) ===") integral1_f1, z1 = complex_line_integral(f1, path1, t_range1) print(f"∫ z dz = {integral1_f1:.6f}") print(f"理論値: (1+i)^2 / 2 = {(1+1j)**2 / 2:.6f}") integral1_f2, _ = complex_line_integral(f2, path1, t_range1) print(f"\n∫ z^2 dz = {integral1_f2:.6f}") print(f"理論値: (1+i)^3 / 3 = {(1+1j)**3 / 3:.6f}") # 経路2: 単位円(反時計回り) path2 = lambda t: np.exp(1j * t) t_range2 = (0, 2*np.pi) print("\n\n=== 経路2: 単位円(反時計回り) ===") integral2_f1, z2 = complex_line_integral(f1, path2, t_range2) print(f"∫ z dz = {integral2_f1:.6f}") print(f"理論値: 0 (正則関数の閉曲線積分)") integral2_f3, _ = complex_line_integral(f3, path2, t_range2) print(f"\n∫ (1/z) dz = {integral2_f3:.6f}") print(f"理論値: 2πi = {2*np.pi*1j:.6f}") # 経路3: 原点から1+iへの折れ線 (0→1→1+i) path3a = lambda t: t # 0 → 1 (実軸) path3b = lambda t: 1 + 1j*t # 1 → 1+i (虚軸) integral3_f1a, z3a = complex_line_integral(f1, path3a, (0, 1)) integral3_f1b, z3b = complex_line_integral(f1, path3b, (0, 1)) integral3_f1_total = integral3_f1a + integral3_f1b print("\n\n=== 経路3: 折れ線 (0→1→1+i) ===") print(f"∫ z dz = {integral3_f1_total:.6f}") print(f"理論値: (1+i)^2 / 2 = {(1+1j)**2 / 2:.6f}") print(f"(経路によらない = 正則関数)") # 可視化 fig, axes = plt.subplots(1, 3, figsize=(16, 5)) # 経路1 ax = axes[0] ax.plot(z1.real, z1.imag, 'b-', linewidth=2, label='経路') ax.plot(0, 0, 'go', markersize=10, label='始点') ax.plot(1, 1, 'ro', markersize=10, label='終点') ax.arrow(0, 0, 0.4, 0.4, head_width=0.05, head_length=0.05, fc='blue', ec='blue') ax.grid(True, alpha=0.3) ax.set_xlabel('Re(z)', fontsize=12) ax.set_ylabel('Im(z)', fontsize=12) ax.set_title('経路1: 直線 $(0 \\to 1+i)$', fontsize=12) ax.axis('equal') ax.legend() # 経路2 ax = axes[1] ax.plot(z2.real, z2.imag, 'b-', linewidth=2, label='経路') ax.plot(1, 0, 'go', markersize=10, label='始点/終点') # 矢印(反時計回り) angles_arrow = [0, np.pi/2, np.pi, 3*np.pi/2] for angle in angles_arrow: z_arr = np.exp(1j * angle) dz = 0.3 * np.exp(1j * (angle + np.pi/2)) ax.arrow(z_arr.real, z_arr.imag, dz.real, dz.imag, head_width=0.1, head_length=0.08, fc='blue', ec='blue', alpha=0.6) ax.grid(True, alpha=0.3) ax.set_xlabel('Re(z)', fontsize=12) ax.set_ylabel('Im(z)', fontsize=12) ax.set_title('経路2: 単位円(反時計回り)', fontsize=12) ax.axis('equal') ax.legend() # 経路3 ax = axes[2] ax.plot(z3a.real, z3a.imag, 'b-', linewidth=2, label='経路') ax.plot(z3b.real, z3b.imag, 'b-', linewidth=2) ax.plot(0, 0, 'go', markersize=10, label='始点') ax.plot(1, 1, 'ro', markersize=10, label='終点') ax.plot(1, 0, 'ko', markersize=8, label='中間点') ax.arrow(0, 0, 0.4, 0, head_width=0.05, head_length=0.05, fc='blue', ec='blue') ax.arrow(1, 0, 0, 0.4, head_width=0.05, head_length=0.05, fc='blue', ec='blue') ax.grid(True, alpha=0.3) ax.set_xlabel('Re(z)', fontsize=12) ax.set_ylabel('Im(z)', fontsize=12) ax.set_title('経路3: 折れ線 $(0 \\to 1 \\to 1+i)$', fontsize=12) ax.axis('equal') ax.legend() plt.suptitle('複素積分の経路', fontsize=14) plt.tight_layout() plt.show()

import numpy as np import matplotlib.pyplot as plt def complex_contour_integral(f, path, t_range, n_points=1000): """閉曲線に沿った複素積分""" t = np.linspace(t_range[0], t_range[1], n_points) dt = t[1] - t[0] z = path(t) dz_dt = np.gradient(z, dt) f_z = f(z) integral = np.trapz(f_z * dz_dt, dx=dt) return integral, z # 正則関数 f_analytic = [ ('z', lambda z: z), ('z^2', lambda z: z**2), ('e^z', lambda z: np.exp(z)), ('sin(z)', lambda z: np.sin(z)), ] # 非正則関数 f_not_analytic = [ ('z̄', lambda z: np.conj(z)), ('|z|^2', lambda z: np.abs(z)**2), ] # 特異点を含む関数 f_singular = [ ('1/z', lambda z: 1/z), # z=0 で特異点 ('1/(z-1)', lambda z: 1/(z-1)), # z=1 で特異点 ] # 閉曲線: 単位円 unit_circle = lambda t: np.exp(1j * t) t_range = (0, 2*np.pi) print("=== コーシーの積分定理の検証 ===\n") print("閉曲線: 単位円(反時計回り)\n") print("【正則関数(特異点なし)】") print("コーシーの積分定理により ∮f(z)dz = 0 のはず:") for name, f in f_analytic: integral, _ = complex_contour_integral(f, unit_circle, t_range) print(f" ∮ {name:8s} dz = {integral.real:10.6f} + {integral.imag:10.6f}i") print("\n【非正則関数】") print("一般に ∮f(z)dz ≠ 0:") for name, f in f_not_analytic: integral, _ = complex_contour_integral(f, unit_circle, t_range) print(f" ∮ {name:8s} dz = {integral.real:10.6f} + {integral.imag:10.6f}i") print("\n【特異点を含む関数】") print("単位円内に特異点 → ∮f(z)dz ≠ 0:") for name, f in f_singular: integral, _ = complex_contour_integral(f, unit_circle, t_range) print(f" ∮ {name:12s} dz = {integral.real:10.6f} + {integral.imag:10.6f}i") # 可視化: 様々な閉曲線での検証 contours = { '単位円': (lambda t: np.exp(1j * t), (0, 2*np.pi)), '正方形': (lambda t: np.where(t < 0.25, 4*t, np.where(t < 0.5, 1 + 4j*(t-0.25), np.where(t < 0.75, 1 - 4*(t-0.5) + 1j, 4j*(1 - (t-0.75))))), (0, 1)), '楕円': (lambda t: 2*np.cos(t) + 1j*np.sin(t), (0, 2*np.pi)), } f_test = lambda z: z**2 # 正則関数 fig, axes = plt.subplots(1, 3, figsize=(15, 5)) for ax, (name, (path, t_range)) in zip(axes, contours.items()): integral, z_path = complex_contour_integral(f_test, path, t_range) ax.plot(z_path.real, z_path.imag, 'b-', linewidth=2, label=f'経路: {name}') ax.plot(z_path.real[0], z_path.imag[0], 'go', markersize=10, label='始点') ax.grid(True, alpha=0.3) ax.axhline(0, color='gray', linewidth=0.5) ax.axvline(0, color='gray', linewidth=0.5) ax.set_xlabel('Re(z)', fontsize=12) ax.set_ylabel('Im(z)', fontsize=12) ax.set_title(f'{name}\n∮ $z^2$ dz = {integral:.6f}', fontsize=12) ax.axis('equal') ax.legend() plt.suptitle('コーシーの積分定理: 正則関数の閉曲線積分 = 0', fontsize=14) plt.tight_layout() plt.show()

import numpy as np import matplotlib.pyplot as plt def cauchy_integral_formula(f, z0, radius=1.0, n_points=1000): """ コーシーの積分公式を使ってf(z0)を計算 f: 正則関数 z0: 評価点 radius: z0を中心とする円の半径 """ t = np.linspace(0, 2*np.pi, n_points) dt = t[1] - t[0] # 経路: z0を中心とする円 z = z0 + radius * np.exp(1j * t) dz_dt = 1j * radius * np.exp(1j * t) # 被積分関数 integrand = f(z) / (z - z0) # 積分 integral = np.trapz(integrand * dz_dt, dx=dt) f_z0_from_integral = integral / (2 * np.pi * 1j) return f_z0_from_integral, z # テスト関数 functions = { 'z^2': lambda z: z**2, 'z^3': lambda z: z**3, 'e^z': lambda z: np.exp(z), 'sin(z)': lambda z: np.sin(z), } # 評価点 z0 = 0.5 + 0.8j print("=== コーシーの積分公式の検証 ===") print(f"評価点: z0 = {z0}\n") results = [] for name, f in functions.items(): # 直接計算 f_z0_direct = f(z0) # コーシーの積分公式で計算 f_z0_integral, path = cauchy_integral_formula(f, z0, radius=1.0) error = np.abs(f_z0_direct - f_z0_integral) print(f"f(z) = {name}") print(f" 直接計算: f(z0) = {f_z0_direct:.8f}") print(f" 積分公式から: f(z0) = {f_z0_integral:.8f}") print(f" 誤差: {error:.2e}\n") results.append((name, f, path)) # 可視化 fig, axes = plt.subplots(2, 2, figsize=(12, 12)) axes = axes.flatten() for idx, (name, f, path) in enumerate(results): if idx >= len(axes): break ax = axes[idx] # 関数の等高線 x = np.linspace(-2, 2, 100) y = np.linspace(-2, 2, 100) X, Y = np.meshgrid(x, y) Z = X + 1j*Y W = f(Z) im = ax.contourf(X, Y, np.abs(W), levels=20, cmap='viridis', alpha=0.6) plt.colorbar(im, ax=ax, label='|f(z)|') # 積分経路 ax.plot(path.real, path.imag, 'r-', linewidth=2, label='積分経路') ax.plot(z0.real, z0.imag, 'ko', markersize=10, label=f'z0 = {z0:.2f}') ax.grid(True, alpha=0.3) ax.set_xlabel('Re(z)', fontsize=12) ax.set_ylabel('Im(z)', fontsize=12) ax.set_title(f'$f(z) = {name}$', fontsize=12) ax.axis('equal') ax.legend() ax.set_xlim(-1.5, 2) ax.set_ylim(-1, 2) plt.suptitle('コーシーの積分公式: 円周上の値から中心の値を計算', fontsize=14) plt.tight_layout() plt.show()

import numpy as np import matplotlib.pyplot as plt from scipy.special import factorial def cauchy_derivative_formula(f, z0, n, radius=1.0, n_points=1000): """ コーシーの導関数公式を使ってf^(n)(z0)を計算 f: 正則関数 z0: 評価点 n: 導関数の階数 """ t = np.linspace(0, 2*np.pi, n_points) dt = t[1] - t[0] # 経路: z0を中心とする円 z = z0 + radius * np.exp(1j * t) dz_dt = 1j * radius * np.exp(1j * t) # 被積分関数 integrand = f(z) / (z - z0)**(n + 1) # 積分 integral = np.trapz(integrand * dz_dt, dx=dt) f_n_z0 = factorial(n) * integral / (2 * np.pi * 1j) return f_n_z0 # 複素微分の数値計算(検証用) def numerical_derivative(f, z, n=1, h=1e-5): """n階微分の数値計算""" if n == 0: return f(z) elif n == 1: return (f(z + h) - f(z - h)) / (2*h) else: # 再帰的に計算 f_n_minus_1 = lambda w: numerical_derivative(f, w, n-1, h) return (f_n_minus_1(z + h) - f_n_minus_1(z - h)) / (2*h) # テスト関数 f = lambda z: np.exp(z) z0 = 1.0 + 0.5j print("=== 高階導関数の積分公式 ===") print(f"f(z) = e^z, z0 = {z0}\n") print("理論値: すべての導関数が e^z0\n") for n in range(5): # 積分公式 f_n_integral = cauchy_derivative_formula(f, z0, n, radius=0.5) # 数値微分 f_n_numerical = numerical_derivative(f, z0, n) # 理論値 (e^z の n 階導関数は e^z) f_n_theory = np.exp(z0) error_integral = np.abs(f_n_integral - f_n_theory) error_numerical = np.abs(f_n_numerical - f_n_theory) print(f"f^({n})(z0):") print(f" 積分公式: {f_n_integral:.8f}") print(f" 数値微分: {f_n_numerical:.8f}") print(f" 理論値: {f_n_theory:.8f}") print(f" 誤差(積分): {error_integral:.2e}") print(f" 誤差(数値): {error_numerical:.2e}\n") # 別のテスト: f(z) = z^4 print("\n=== f(z) = z^4 ===") f2 = lambda z: z**4 z0_2 = 2.0 + 1.0j derivatives_theory = [ z0_2**4, # f 4 * z0_2**3, # f' 12 * z0_2**2, # f'' 24 * z0_2, # f''' 24, # f'''' ] for n in range(5): f_n_integral = cauchy_derivative_formula(f2, z0_2, n, radius=1.0) f_n_theory = derivatives_theory[n] error = np.abs(f_n_integral - f_n_theory) print(f"f^({n})(z0) = {f_n_integral:.6f}, 理論値 = {f_n_theory:.6f}, 誤差 = {error:.2e}") # 可視化: 異なる半径での精度 radii = np.linspace(0.1, 2.0, 20) errors = [] for radius in radii: f_2_integral = cauchy_derivative_formula(f2, z0_2, 2, radius=radius) error = np.abs(f_2_integral - derivatives_theory[2]) errors.append(error) plt.figure(figsize=(10, 6)) plt.semilogy(radii, errors, 'b-o', linewidth=2, markersize=6) plt.grid(True, alpha=0.3) plt.xlabel('積分経路の半径', fontsize=12) plt.ylabel('誤差', fontsize=12) plt.title("コーシーの導関数公式の精度 vs 積分経路の半径\n$f''(z_0)$ for $f(z) = z^4$", fontsize=14) plt.axhline(1e-10, color='red', linestyle='--', label='数値誤差の限界') plt.legend() plt.tight_layout() plt.show()

import numpy as np import matplotlib.pyplot as plt def plot_conformal_map(f, title): """等角写像の可視化""" # 元の領域: 格子 x = np.linspace(-1.5, 1.5, 15) y = np.linspace(-1.5, 1.5, 15) fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6)) # 左図: 元の領域 for xi in x: y_line = y z_line = xi + 1j*y_line ax1.plot(z_line.real, z_line.imag, 'b-', linewidth=1, alpha=0.5) for yi in y: x_line = x z_line = x_line + 1j*yi ax1.plot(z_line.real, z_line.imag, 'b-', linewidth=1, alpha=0.5) ax1.axhline(0, color='gray', linewidth=0.5) ax1.axvline(0, color='gray', linewidth=0.5) ax1.grid(True, alpha=0.3) ax1.set_xlabel('Re(z)', fontsize=12) ax1.set_ylabel('Im(z)', fontsize=12) ax1.set_title('元の領域 (z平面)', fontsize=12) ax1.axis('equal') ax1.set_xlim(-2, 2) ax1.set_ylim(-2, 2) # 右図: 写像後の領域 for xi in x: y_line = y z_line = xi + 1j*y_line w_line = f(z_line) ax2.plot(w_line.real, w_line.imag, 'r-', linewidth=1, alpha=0.5) for yi in y: x_line = x z_line = x_line + 1j*yi w_line = f(z_line) ax2.plot(w_line.real, w_line.imag, 'r-', linewidth=1, alpha=0.5) ax2.axhline(0, color='gray', linewidth=0.5) ax2.axvline(0, color='gray', linewidth=0.5) ax2.grid(True, alpha=0.3) ax2.set_xlabel('Re(w)', fontsize=12) ax2.set_ylabel('Im(w)', fontsize=12) ax2.set_title(f'写像後の領域 (w平面): {title}', fontsize=12) ax2.axis('equal') plt.tight_layout() return fig # 様々な等角写像 maps = { 'w = z^2': lambda z: z**2, 'w = e^z': lambda z: np.exp(z), 'w = 1/z': lambda z: 1/z, 'w = (z-1)/(z+1) (Möbius)': lambda z: (z-1)/(z+1), } for name, f in maps.items(): fig = plot_conformal_map(f, name) plt.show() print("等角写像の性質:") print("- 格子の交点での角度が保たれる") print("- 正則関数は局所的に回転と拡大の合成")

import numpy as np import matplotlib.pyplot as plt def laplacian_2d(u, x, y, h=0.01): """2次元ラプラシアンの数値計算""" u_xx = (u(x+h, y) - 2*u(x, y) + u(x-h, y)) / h**2 u_yy = (u(x, y+h) - 2*u(x, y) + u(x, y-h)) / h**2 return u_xx + u_yy # 正則関数 f(z) = z^2 = (x+iy)^2 = (x^2-y^2) + i(2xy) def u(x, y): """実部: u = x^2 - y^2""" return x**2 - y**2 def v(x, y): """虚部: v = 2xy""" return 2*x*y # グリッド x = np.linspace(-2, 2, 100) y = np.linspace(-2, 2, 100) X, Y = np.meshgrid(x, y) U = u(X, Y) V = v(X, Y) # ラプラシアンの計算(数値微分) dx = x[1] - x[0] dy = y[1] - y[0] Laplacian_U = (np.roll(U, -1, axis=1) - 2*U + np.roll(U, 1, axis=1)) / dx**2 + \ (np.roll(U, -1, axis=0) - 2*U + np.roll(U, 1, axis=0)) / dy**2 Laplacian_V = (np.roll(V, -1, axis=1) - 2*V + np.roll(V, 1, axis=1)) / dx**2 + \ (np.roll(V, -1, axis=0) - 2*V + np.roll(V, 1, axis=0)) / dy**2 # テスト点でのラプラシアン x0, y0 = 1.0, 1.5 lap_u = laplacian_2d(u, x0, y0) lap_v = laplacian_2d(v, x0, y0) print("=== 調和関数の検証 ===") print(f"正則関数: f(z) = z^2") print(f"実部: u(x,y) = x^2 - y^2") print(f"虚部: v(x,y) = 2xy") print(f"\nテスト点: ({x0}, {y0})") print(f"∇²u = {lap_u:.6e} (should be ~0)") print(f"∇²v = {lap_v:.6e} (should be ~0)") # 可視化 fig, axes = plt.subplots(2, 3, figsize=(16, 10)) # 実部 im = axes[0, 0].contourf(X, Y, U, levels=20, cmap='RdBu_r') axes[0, 0].contour(X, Y, U, levels=10, colors='black', linewidths=0.5, alpha=0.4) axes[0, 0].set_xlabel('x', fontsize=12) axes[0, 0].set_ylabel('y', fontsize=12) axes[0, 0].set_title('実部: $u = x^2 - y^2$', fontsize=12) axes[0, 0].axis('equal') plt.colorbar(im, ax=axes[0, 0]) # 虚部 im = axes[0, 1].contourf(X, Y, V, levels=20, cmap='RdBu_r') axes[0, 1].contour(X, Y, V, levels=10, colors='black', linewidths=0.5, alpha=0.4) axes[0, 1].set_xlabel('x', fontsize=12) axes[0, 1].set_ylabel('y', fontsize=12) axes[0, 1].set_title('虚部: $v = 2xy$', fontsize=12) axes[0, 1].axis('equal') plt.colorbar(im, ax=axes[0, 1]) # 実部と虚部の等高線(直交性) axes[0, 2].contour(X, Y, U, levels=15, colors='blue', linewidths=1.5, alpha=0.7) axes[0, 2].contour(X, Y, V, levels=15, colors='red', linewidths=1.5, alpha=0.7) axes[0, 2].set_xlabel('x', fontsize=12) axes[0, 2].set_ylabel('y', fontsize=12) axes[0, 2].set_title('等高線の直交性(青:u, 赤:v)', fontsize=12) axes[0, 2].axis('equal') axes[0, 2].grid(True, alpha=0.3) # ラプラシアン(実部) im = axes[1, 0].contourf(X, Y, Laplacian_U, levels=20, cmap='seismic', vmin=-0.1, vmax=0.1) axes[1, 0].set_xlabel('x', fontsize=12) axes[1, 0].set_ylabel('y', fontsize=12) axes[1, 0].set_title('$\\nabla^2 u$ (should be ~0)', fontsize=12) axes[1, 0].axis('equal') plt.colorbar(im, ax=axes[1, 0]) # ラプラシアン(虚部) im = axes[1, 1].contourf(X, Y, Laplacian_V, levels=20, cmap='seismic', vmin=-0.1, vmax=0.1) axes[1, 1].set_xlabel('x', fontsize=12) axes[1, 1].set_ylabel('y', fontsize=12) axes[1, 1].set_title('$\\nabla^2 v$ (should be ~0)', fontsize=12) axes[1, 1].axis('equal') plt.colorbar(im, ax=axes[1, 1]) # ラプラシアンの大きさ Lap_magnitude = np.sqrt(Laplacian_U**2 + Laplacian_V**2) im = axes[1, 2].contourf(X, Y, Lap_magnitude, levels=20, cmap='viridis') axes[1, 2].set_xlabel('x', fontsize=12) axes[1, 2].set_ylabel('y', fontsize=12) axes[1, 2].set_title('$|\\nabla^2 u|^2 + |\\nabla^2 v|^2$', fontsize=12) axes[1, 2].axis('equal') plt.colorbar(im, ax=axes[1, 2]) plt.suptitle('調和関数: 正則関数の実部・虚部はラプラス方程式を満たす', fontsize=14) plt.tight_layout() plt.show() print("\n調和関数の性質:") print("- 正則関数の実部と虚部は調和関数") print("- 調和関数の等高線は直交する(コーシー・リーマンの方程式)") print("- 調和関数は極大値・極小値を持たない(最大値原理)")