By V.I. Fabrikani

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**Additional resources for Applications of potential theory in mechanics. Selection of new results**

**Example text**

34) can be expressed in elementary functions (Bateman and Erdelyi, 1955) F( 1 1 3 (1 − ζ)n+1/2 dn ζn-1/2 sin-1√ζ. − n , ; ; ζ) = 2 2 2 Γ( n + 1) dζn √1 − ζ Example 2. Let the charge distribution be prescribed σ(ρ,φ)=σnρncosn φ, σn=const. 4 Internal mixed boundary value problem for a half-space 2 n + 3 dn 1 1 1 + √ζ 5 3 . 37) Bibliographical note. The degree of effectiveness of the new approach can be established by comparison with the results reported in the literature. 21), can be found in the works of Lord Kelvin.

15 v ( a 2 − l 21)1/2 3 1 e3 i φ 1 − Answer: V (ρ,φ, z ) = 2 a 4ρ3 ( a 2 − l 21)1/2 3 ( a 2 − l 21)1/2 5 1 1 + 1 − . 1 − − 3 10 a a 8. Let the following boundary conditions be prescribed at z =0: for ρ≤ a , V = 0, 0≤φ<2π; ∂V = − 2πσ /ρ2, σ =const, for ρ> a , 0≤φ<2π. 0 0 ∂z Find the potential function and the charge distribution. 2πσ l [(ρ2 + z 2)1/2 + ( l 22 − a 2)1/2] 0 2 , Answer: V (ρ,φ, z ) = 2 2 1/2 ln (ρ + z ) a [(ρ2 + z 2)1/2 + z ] σ σ a , σ(ρ,φ) = 2 ℜ1 − 2 ( a − ρ2)1/2 ρ 0 9.

Two examples are considered below. Example 1. Let the potential prescribed inside the disc be v (ρ,φ)= v nρncosn φ, v n=const. 16) is l 2vn 1 x 2nd x Γ( n + 1) V (ρ,φ, z ) = cosn φ ⌠ 2 2 1/2 √π ρn Γ( n + 1) ⌡ (ρ − x ) 0 2 2 2 1/2 l2 − a2 2Γ( n + 1) ( l 2 − a ) 1 1 3 2 . 34) can be expressed in elementary functions (Bateman and Erdelyi, 1955) F( 1 1 3 (1 − ζ)n+1/2 dn ζn-1/2 sin-1√ζ. − n , ; ; ζ) = 2 2 2 Γ( n + 1) dζn √1 − ζ Example 2. Let the charge distribution be prescribed σ(ρ,φ)=σnρncosn φ, σn=const.

### Applications of potential theory in mechanics. Selection of new results by V.I. Fabrikani

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