经典电动力学导论 Let there be light 第一章:数学基础§1.6 维 Dirac delta函数 6(x-a) ≠ 且 a+n a)dc 6(x-a)dx=1 6(x)称为 Dirac delta函数 数学上称为广义函数( generalized function)或分布( distribution) 广义函数专著: Lighthill“ Fourier Analysis and generalized Functions"( Cambridge University1964) 复旦大学物理系 林志方徐建军3
Let there be light ²;>ÄåÆØ 1ÙµêÆÄ: § 1.6 ! Dirac delta ¼ê δ(x − a) = 0 x 6= a ∞ x = a Z +∞ −∞ δ(x − a) dx = Z a+η a−η δ(x − a) dx = 1 δ(x) ¡ Dirac delta ¼ê êÆþ¡2¼ê (generalized function) ½©Ù (distribution) 2¼ê;͵Lighthill “Fourier Analysis and Generalized Functions” (Cambridge University 1964) EÆ ÔnX Mï 3
经典电动力学导论 Let there be light 第一章:数学基础§1.6 维 Dirac delta函数 6(x-a) ≠ 且 a+n a)dc 6(x-a)dx=1 6(x)称为 Dirac delta函数 数学上称为广义函数( generalized function)或分布( distribution) 广义函数专著: Lighthill" Fourier Analysis and Generalized Functions"( Cambridge University1964) 对任意平滑函数,由于f(x)6(x-a)=f(a)6(x-a) 复旦大学物理系 林志方徐建军3
Let there be light ²;>ÄåÆØ 1ÙµêÆÄ: § 1.6 ! Dirac delta ¼ê δ(x − a) = 0 x 6= a ∞ x = a Z +∞ −∞ δ(x − a) dx = Z a+η a−η δ(x − a) dx = 1 δ(x) ¡ Dirac delta ¼ê êÆþ¡2¼ê (generalized function) ½©Ù (distribution) 2¼ê;͵Lighthill “Fourier Analysis and Generalized Functions” (Cambridge University 1964) é?¿²w¼ê§du f(x) δ(x − a) = f(a) δ(x − a) EÆ ÔnX Mï 3
经典电动力学导论 Let there be light 第一章:数学基础§1.6 维 Dirac delta函数 6(x-a) ≠ 且 a+n a)dc 6(x-a)dx=1 6(x)称为 Dirac delta函数 数学上称为广义函数( generalized function)或分布( distribution) 广义函数专著: Lighthill" Fourier Analysis and Generalized Functions"( Cambridge University1964) 对任意平滑函数,由于f(x)6(x-a)=f(a)6(x-a),故有 复旦大学物理系 林志方徐建军3
Let there be light ²;>ÄåÆØ 1ÙµêÆÄ: § 1.6 ! Dirac delta ¼ê δ(x − a) = 0 x 6= a ∞ x = a Z +∞ −∞ δ(x − a) dx = Z a+η a−η δ(x − a) dx = 1 δ(x) ¡ Dirac delta ¼ê êÆþ¡2¼ê (generalized function) ½©Ù (distribution) 2¼ê;͵Lighthill “Fourier Analysis and Generalized Functions” (Cambridge University 1964) é?¿²w¼ê§du f(x) δ(x − a) = f(a) δ(x − a)§k EÆ ÔnX Mï 3
经典电动力学导论 Let there be light 第一章:数学基础§1.6 维 Dirac delta函数 6(x-a) ≠ 且 a+n 6(x-a)dx=1 6(x)称为 Dirac delta函数 数学上称为广义函数( generalized function)或分布( distribution) 广义函数专著: Lighthill" Fourier Analysis and Generalized Functions"( Cambridge University1964) 对任意平滑函数,由于f(x)6(x-a)=f(a)6(x-a),故有 atn atn f()8(a-a)d ao(r-a ar f(a d()d c=f(a 复旦大学物理系 林志方徐建军3
Let there be light ²;>ÄåÆØ 1ÙµêÆÄ: § 1.6 ! Dirac delta ¼ê δ(x − a) = 0 x 6= a ∞ x = a Z +∞ −∞ δ(x − a) dx = Z a+η a−η δ(x − a) dx = 1 δ(x) ¡ Dirac delta ¼ê êÆþ¡2¼ê (generalized function) ½©Ù (distribution) 2¼ê;͵Lighthill “Fourier Analysis and Generalized Functions” (Cambridge University 1964) é?¿²w¼ê§du f(x) δ(x − a) = f(a) δ(x − a)§k Z +∞ −∞ f(x)δ(x−a) dx = Z a+η a−η f(a)δ(x−a) dx = f(a) Z a+η a−η δ(x−a) dx = f(a) EÆ ÔnX Mï 3
经典电动力学导论 Let there be light 第一章:数学基础§1.6 δ函数的性质 1.0(-x)=6( o(C 4.6(kx) 5.f(x)06(x-a)=f(a)6(x-a r6(x)=0 6.0(x)=∑ (a-ci ,函数g(x)只有单重根,∑对g(x)的所有单重根求和 7.6(x)=0 1. if a>0 dr 6(x)= 0. if <0 阶跃函数( step function) 复旦大学物理系 林志方徐建军4
Let there be light ²;>ÄåÆØ 1ÙµêÆÄ: § 1.6 δ ¼ê5 1. δ(−x) = δ(x) 2. δ 0 (−x) = −δ 0 (x) 3. xδ0 (x) = −δ(x) 4. δ(kx) = 1 |k| δ(x) 5. f(x)δ(x − a) = f(a)δ(x − a)§ xδ(x) = 0 6. δ [g(x)] = X i δ(x − xi) |g 0(xi)| , ¼ê g(x) kü§ X i é g(x) ¤kü¦Ú 7. δ(x) = dθ(x) d x , θ(x) = 1, if x ≥ 0 0, if x < 0 ¼ê (step function) EÆ ÔnX Mï 4