發問時間: 科學數學 · 1 0 年前

複變解題HARMONIC 在D的定義?

1.suppose that the functions u and v in a domain D

Is the product uv nessarily harmonic in D?

2.HARMONIC 在Dㄉ定義?

3.HARMONIC 在D(很多力提都會說v is harmonic conjugate of u或v is harmonic conjugate for u或 沒說of,for, 三種是一樣ㄉ嗎?)

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  • 1 0 年前
    最佳解答

    1.suppose that the functions u and v in a domain D

    Is the product uv nessarily harmonic in D?

    Do you mean suppose that the functions u and v are harmonic in a domain D

    Is the product uv nessarily harmonic in D?

    No.

    2. HARMONIC 在Dㄉ定義?

    A function of two variabls u=u(x,y) is said to be harmonic in a domain D if it satisfies the Laplace equation: u_xx+u_yy=0 for all point(x,y) in D.

    3. conjugate (共軛 ) 必須成對提及 故說 v is a harmonic conjugate of u , that means f(z)=f(x+iy)=u(x,u)+iv(x,y) is an analytic function.--- v is conjugate to u. 如定義1, 若討論 harmonic only, 無需 of.

    2010-01-11 02:38:53 補充:

    A counterexample for 1.:

    u(x,y)=v(x,y)=xy are harmonic functions [please check they satisfy Laplace equation]. The product u by v=x^2*y^2, say w(x,y). Then w_xx=2y^2, and w_yy=2x^2,. Obviously w_xx+w_yy not equal to 0!

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