# 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?)

1.請說為什麼?

### 1 個解答

• 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!