By Vladislav V. Kravchenko
Pseudoanalytic functionality conception generalizes and preserves many an important beneficial properties of complicated analytic functionality thought. The Cauchy-Riemann process is changed by means of a way more common first-order approach with variable coefficients which seems to be heavily regarding very important equations of mathematical physics. This relation provides strong instruments for learning and fixing Schr?dinger, Dirac, Maxwell, Klein-Gordon and different equations through complex-analytic equipment. The publication is devoted to those fresh advancements in pseudoanalytic functionality conception and their functions in addition to to multidimensional generalizations. it's directed to undergraduates, graduate scholars and researchers drawn to complex-analytic tools, resolution concepts for equations of mathematical physics, partial and traditional differential equations.
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Extra info for Applied Pseudoanalytic Function Theory (Frontiers in Mathematics)
5) is zero. 17) and hence the real-valued function ϕ = A[w/f ] is well deﬁned and satisﬁes the equation ∂z ϕ = w/f . Consider the expression w +w f w ∂z f w A + (∂z f ) ∂z A − w. 2. Factorization of the operator div p grad +q. 7) where the following observation was used: ∂y A[ x u + iv ](x, y) = 2 f v(x0 , y) u(η, y) dη − f (η, y) f (x0 , y) x v(x0 , y) v(η, y) dη − ∂η f (η, y) f (x0 , y) x0 ∂y x0 = −2 =− 2v(x, y) . 6) gives us the equality w Δ(Sw) = νf A[ ] = νSw. f Proposition 27. 1). Then SP g = g + cf where c is an arbitrary real constant.
An important special case 37 y cy sinh(cy) 2x sinh(cy) e + ,.... , u5 (x, y) = − c c2 c Formal powers of higher order can be constructed explicitly using a computer algebra system. 5 allowed us to obtain analytic expressions for the formal powers up to order ten, which gave us the ﬁrst twenty-one functions u1 , . . , u21 . u4 (x, y) = x2 − The generating pair considered in this example (F, G) = (f (y), i/f (y)) belongs to a more general class of generating pairs for which L. Bers found elegant explicit formulas for corresponding formal powers which we give in the next section.
12) n=0 where the absence of the subindex m means that all the formal powers correspond to the same generating pair (F, G) and the series converges uniformly in some neighborhood of z0 . Based on the similarity principle it can be shown  that the uniform limit of pseudoanalytic functions is pseudoanalytic, and that a uniformly convergent series of (F, G)-pseudoanalytic functions can be (F, G)-diﬀerentiated term by term. 12) is (F, G)-pseudoanalytic and its rth derivative admits the expansion ∞ n(n − 1) · · · (n − r + 1)Zr(n−r) (an , z0 ; z).
Applied Pseudoanalytic Function Theory (Frontiers in Mathematics) by Vladislav V. Kravchenko