R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 68
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. · ) .dg /d\,n= *r/' (Z)] = - * [ } / « - i C Z ) ] = m(n = 1, 2, · · · ) ,1, 2, · · · . L e tThen from E q . ( 3 2 ) , since £ [ / » ( £ ) ] =^/ _ χ ( 0 dt~ ζgff-nand (29) is satisfied. Substituting successively into E q . (32) andfinally21.5365BERGMAN'S INTEGRATION METHODwriting the w-tuple integral as a single integral we obtain(32')f (Z)=n2}~{l} \y[foitKZ-tV-1(n =dt1, 2, · · • ) ,so that, with dfo/dt = f'o ^ 0, and assuming from now on t h a t / ( 0 ) = 0:0(32")=f {Z)n[ fO(t)(ZJo-Z2 nlnt)n{η = 0, 1, · · · ) .dtThis last formula may be verified by applying partial integration to theintegral in (32').
For η =/o(0) = fo(Z).0 the right side of (32") reduces to fo(Z)0 » ( M ) = LJ¥A(33')—Thus, on account of Eq. (33)Z nlnF f (tHZ[_Jo0-t) dtn(n = 0, 1,2, · . . ) ,Substituting these into Eq. (27) we obtain(34)ψ*(\,θ)=Σ ^ τn=0^ W *Δ 711ΓJof'o(t)(Z- t )nd t.T h e similarity of (34) and (23) is obvious.If we introduce the function of three variablesC ( * ; M ) = £J-G„(X)(* - ® >(35)nn—0ΔTl !we can write (34) in the formφ*(λ,θ) =(36)where df (t)a(37)= f' (t)dt.03jfd/ (0],G(t;\6)oAlso, using (25),φ(\,θ) = v$fG(t;\,e)df (t)0•T h e right side of E q . (37) thus transforms the arbitraryfo(Z)analyticfunctioninto a solution ^ ( λ , θ) of the second-order equation ( 6 ) .W i t h respect to the legitimacy of these operations, we note that to justifythe interchange of differentiation and summation which led to (34) weneed the uniform convergence of this series and its derivatives in a λ,0region.
T h e interchange of summation and integration which leads from(34) to (36) is not essential for our final result. T o justify it we need uniform convergence in a ^-region for fixed λ,0. Both are covered by the considerations in the next section.N e x t , let us make an appropriate choice of the arbitrary functionf (t)0V. INTEGRATION THEORY AND SHOCKS366which occurs in these various formulas. Denote by W o ( f ) with f = qe~the hodograph potential of an incompressible problem.
Then $[w (£)]=ψο(ς> Θ) is the incompressible stream function of this problem, and we chooset90(38)/ ( 0 = ιυοίβ'),= 0.w {l)00Then, consider the passage to the limit q —> <» for fixed (q, Θ). For greaterclarity we write ^(q 0) or ψ*(#, 0) instead of ^ ( λ , 0) or ^ * ( λ , 0).
W e assume (it will be proved presently) thatmylim G(t; λ, 0) = 1.<7m-**>( I n a formal way, this is seen from (35) and (29") and Go = 1). N o t e alsothat in (37) V —•> 1 as q —» oo. W e have then from (36) and (37), using(310 and (38)m(39)limt(q,e) =lim * * ( g , 0 ) = o[w (t)]= >Po(q,0),0as was intended.6. ConvergenceW e now sketch the investigation of the uniform convergence of (35) andof its λ,θ-derivatives as far as the subsonic range is concerned. W e considerfunctions Ρ ( λ ) such that for each e < 0, λ < e, C an appropriate constant,and(40)* =C(e-λ) '2we have(41)|F|*ffand>(* = 1,2, · · · ) .W e will write briefly F « ί for the inequalities (41) and we say thatdominates F or that $ is a majorant function for P .If (41) holds we can find majorant functions Ρ ( λ ) for the G , as follows.W e define P (\)byηnnΡή ι+= Ρ η + C(€ -(η = 0, Ι,· · .
)\T Pn2(42)Po=1,Ρ Λ - ο ο )=0.Then,(43)In fact, for η = 0, we have P(?η0«Ρη.= Go, hence Go <3C Po in the sense of theCONVERGENCE21.6367above definition. N o w suppose GP , then from formulas (29'), (41),(42) we see that G +i « P + i .I t is easily verified that Ρ ( λ ) can be given explicitly asnnnnη(42')= η ! μ„(€ - λ ) "Pnηwhere withαϊ = 1 (44)CY,(1 -μ = 1,«, = i + (i -μ ι = μ (η0η +η2CY+ Π + C)(ft + Ι ) "2α )(η + Ι )= μ « ( η + α ϊ ) (η +2- 2.This recurrence formula for the μ coincides with the recurrence formulafor the coefficients of the hypergeometric series H(a\, α , 1; x) and w e maywriteη2(440H(al9α , 1 ; χ) = Σ μ χ\2ηn=0Since neither ai nor a is a negative integer, the series does not terminate.I t converges for | χ \ < 1, diverges for | χ \ > 1, and converges uniformlywith all its derivatives for | χ \ ^ b < 1, where b is an arbitrary positivenumber less than 1.W e can now estimate the G(t\ λ, θ) of E q .
( 3 5 ) . Substituting (42) and(420 into (35), we obtain2G «Σn^oΡ (λ)2~ηη= Σμ η!2- (η^ο"ηηI IΖ η!t r-λΓ 1η€η!S "· (Ι ΓΤ=ΊΓ11 ζ|Ζ -Μ"" *0 ·Thus, using (440 we have(45)G(t; λ, θ) «Η (ai,α , 1;21^ ".Therefore G, as well as all its derivatives with respect to λ and 0, convergesuniformly and absolutely in a region whereThis justifies all transformations previously performed in Sec. 5 and provesthat the φ* as given in (36) satisfies E q . ( 8 ) .308V. I N T E G R A T I O N T H E O R Y A N DSHOCKSN o w let q —> ° ° .
Since the convergence of the series (35) is also uniformwith respect to q we conclude that G(t; λ, Θ) —* 1, as q —* . W e used thisfact in establishing the result (39).mmm0 0Finally, Ave want to express condition (46) in a geometric form. If the integration of G(t; λ, θ) from t = 0 to t = Ζ is along the straight line from0 to Ζ the maximum of | Ζ — t | is | Ζ | and (46) will be satisfied if | Ζ \ <2 I e - λ I, or(46')Λ +θ < 4(e -22λ) .2T h e boundary of this region (see Fig. 140) is a hyperbola which as e —* 0,becomes3A2-θ 2ScA + 4c = 0,2W e see that the larger qmwhere c = σ + log q.mis, i.e., the more the effects of compressibilityvanish, the larger is the part of the A,0-plane which we obtain as regionof validity and as qm—>0 0, the whole log<?, 0-plane is obtained.θF I G .
140. Region of convergence for Bergman's method.14T h e uni-21.7INTEGRAL309TRANSFORMATIONform convergence of (35) and hence of φ* — Σregion, provided f (Z) itself is bounded there.Ggnnis assured in this0T h e above simple proof must be slightly modified for the polytropic F,where the inequalities (41) do not hold directly. This can be done in severalways, conserving the essential idea of the proof and retaining the result(46) with respect to the region of convergence.157. Integral transformationIn the preceding section we constructed a solution of the .stream-functionequation depending on an arbitrary analytic function f (t) in the convenientform of the integral (37) with the generating function G(t; λ, Θ) defined byEq. (35).
I t seems obvious that this particular generator G is not the onlyone by means of which such a solution can be found. W e shall now characterize such generating functions by means of a differential equation andside conditions which they must satisfy.0Following Bergman we consider the partial differential equation(47)= - ^ j L + cv = 0OZi dzL (v)+2where z\ , z are complex variables and c is an analytic function of Zi , z .Let the function of three independent complex variables Κ(ζι,z , t) be ananalytic function of its variables in some suitable domain B. W e assumethat for all t in Β, Κ satisfies (47) and the boundary condition222(48)—dz= 0fort =z.x2Then, w i t h / ( i ) an arbitrary analytic function of t we form(49)υ(ζ, , z ) =ί21z , t)f'(t)K{z,,2dtJaand prove that υ is a solution of Eq.
(47). Indeed, we haveOZiJoza2OZidzt=\oz2Zl2On account of (48) the second term on the right-hand side vanishes. Sincec is independent of t and Κ satisfies (47), we obtain(50)L+^= [ \ £ k+c K) 's{ t ) d t=-Q370V. I N T E G R A T I O NTHEORY ANDSHOCKST o establish the connection between Eqs. (47) and (8) we extend thevariables Λ, 0 of the preceding section into the complex domain and callthese complex variables Α ι , 0i. Then Ζι = Ai — ιθι and z = Αι + iOi are2independent complex variables, anddz~i "2i Mi'2 Mi+/2z) =i F ( A i ) we see that1 / dv.
dv. ~ \22ν2i θ0ι 'c(zi,if we then put for the c in ( 4 7 ) :r2 θΑιdz2If in Ai and 0i the imaginary parts reduce to zero, the previously independent variables z and z reduce to the conjugate complex variablesΖι = Ζ = A — ιθ, as introduced in E q . (30), and z = Ζ = A + i0. I tthen follows from the fact that L (K)= 0 for all Ζι, z in β thatK ( Z , Ζ, i) satisfiesx22+(51)L(K)2=0+0W e next restrict attention to generators(52)lim K(Z,Z,t)uniformly in ί for fixed 5, 0. For the f(t)(53)+ Z W= 0.for which= 1,in (49) we choose, as in (38),= wo(e'),f(t)where w ( f ) denotes a hodograph potential, as in previous sections. If oneputs as an abbreviation dw /dt = w '(t) so that WQ(e)edt = dwoie'), itfollows from the above construction that00(54)u(Z, Ζ)0=ίK(Z,Z, t)dw (e)0Jasatisfies Eq.
( 8 ) , namelyThen it follows from Eqs. (54), (52), and (31/) that:dwo(e )lI t is thus seen that for ψ*(ς, θ) == w ( f ) + constant.0[u(Z, Z ) ] , with w(Z, Z ) defined by (54),21.7(55)371INTEGRAL TRANSFORMATIONlim \p*(q, Θ) =lim \f/{q, Θ) = ^ ( g , B) +constant,0where ψο(?, θ) = 0 [ w ( f ) ] and φ =o7 ψ * as in E q .
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