Sentence examples for vanish continuously from inspiring English sources

We remark that both V ( r, k ) φ k and W ( r, k ) φ k ( k = 1, 2, 3, … ) are a-harmonic on C n and vanish continuously on S n , where V ( r, k ) and W ( r, k ) are the solutions of equation (1.11) with λ = λ ( Ω, k ).

Let h ( z ) be a harmonic function in C + such that h ( z ) vanishes continuously in ∂ C +.

Then it follows that this is the solution of Equation (1.1) in C n and vanishes continuously on ∂ C n.

Then we see that it is an a-harmonic function in C n and vanishes continuously on ∂ C n.

Consider the harmonic function h'(P =h(P -mathbb{PI}_{mathfrak{C}_{n}(Gamma)}[h](P -mathbb{PI}_{mathfrak{C}_{nly on (mathfrak{S}_{n}(Gamma)) by Lemma 2.

Lemma 2.11 Let H ( r, Θ ) be an a-harmonic function in C n vanishing continuously on ∂ C n , and p, q be two positive integers.

Let (h r,Theta)) be a harmonic function on (mathfrak{C}_{n}(Gamma)) vanishing continuously on (mathfrak {S}_{n}(Gamma)), then (h r,Theta)=mathscr{U}_{h}r^{aleph^varphi(Theta)) for (0< r

If u ∈ F ( p, ρ, α ), then we have u ( z ) = U [ ρ ( | t | ) + α ] ( u ) ( z ) + Im Π ( z ) for all z ∈ C ¯ +, where Π ( z ) is an entire function in C + and vanishes continuously in ∂ C +.

Thus u ( z ) = U [ ρ ( | t | ) + α ] ( u ) ( z ) + Im Π ( z ) for all z ∈ C ¯ +, where Π ( z ) is an entire function in C + and vanishes continuously in ∂ C +. Then we complete the proof of Theorem 2.

Thus u ( z ) = U [ ρ ( | t | ) + β ] ( u ) ( z ) + Im Π ( z ) for all z ∈ C ¯ +, where Π ( z ) is an entire function in C + and vanishes continuously in ∂ C +. Thus we complete the proof of Theorem 4.

Then it follows from Corollary 1 that this is harmonic in C + and vanishes continuously in ∂ C +. Since 0 ≤ ( u ( z ) − U m ( u ) ( z ) ) + ≤ u + ( z ) + U m ( u ) − ( z ) (6.1).