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In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and ), and in signal processing (to describe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger. The most general identity is given by: where is the -th Bessel function of the first kind and is the imaginary unit, Substituting by , we also get: Using the relation valid for integer , the expansion becomes:

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  • Jacobi–Anger expansion (en)
  • 야코비-앙거 전개 (ko)
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  • In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and ), and in signal processing (to describe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger. The most general identity is given by: where is the -th Bessel function of the first kind and is the imaginary unit, Substituting by , we also get: Using the relation valid for integer , the expansion becomes: (en)
  • 야코비–앙거 전개 (또는 야코비–앙거 등식)는 삼각 함수의 지수 꼴을 조화 함수로 풀어 쓰는 것을 말한다. 물리에서 와 사이의 전환 시에, 또는 신호 처리에서 주파수 변조(FM) 신호를 서술할 때 쓰인다. 19세기의 수학자 카를 구스타프 야코프 야코비와 의 이름을 딴 것이다. 가장 일반적인 꼴은 및 이고, 여기서 는 n차 베셀 함수이다. 정수 n에 대해 인 관계를 쓰면 위의 식은 로 다시 쓸 수 있다. 다음과 같은 꼴도 자주 쓰인다. (ko)
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  • In mathematics, the Jacobi–Anger expansion (or Jacobi–Anger identity) is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and ), and in signal processing (to describe FM signals). This identity is named after the 19th-century mathematicians Carl Jacobi and Carl Theodor Anger. The most general identity is given by: where is the -th Bessel function of the first kind and is the imaginary unit, Substituting by , we also get: Using the relation valid for integer , the expansion becomes: (en)
  • 야코비–앙거 전개 (또는 야코비–앙거 등식)는 삼각 함수의 지수 꼴을 조화 함수로 풀어 쓰는 것을 말한다. 물리에서 와 사이의 전환 시에, 또는 신호 처리에서 주파수 변조(FM) 신호를 서술할 때 쓰인다. 19세기의 수학자 카를 구스타프 야코프 야코비와 의 이름을 딴 것이다. 가장 일반적인 꼴은 및 이고, 여기서 는 n차 베셀 함수이다. 정수 n에 대해 인 관계를 쓰면 위의 식은 로 다시 쓸 수 있다. 다음과 같은 꼴도 자주 쓰인다. (ko)
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