Abstract The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S1($ Λ_{0} $, $ Φ_{0} $, Λ, Φ) ofBox 0.1 (R = R0) andV1($ Λ_{0} $, $ Φ_{0} $, Λ, Φ) ofBox 0.2 (R = R0) which depend on theevaluation point {$ Λ_{0} $, $ Φ_{0} $} ∈ SR02 and thesampling point {Λ, Φ} ∈ SR02 ofgravity anomalies $ Δ_{γ} $(Λ, Φ) with respect to a normal gravitational field of typegm/R (”free air anomaly”). If the evaluation point is taken as the meta-north pole of theStokes reference sphere SR02, theStokes function, and theVening-Meinesz function, respectively, takes the formS(Ψ) ofBox 0.1, andV2(Ψ) ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, Ψ} ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {Λ, Φ} onto the tangent plane $ T_{0} $SR02 at {$ Λ_{0} $, $ Φ_{0} $} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S2(Ψ),S3(r),⋯,S6(r) as well as the correspondingStokes-Helmert functions H2(Ψ),H3(r),⋯,H6(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V2(Ψ),V3(r),⋯,V6(r) as well as the correspondingVening-Meinesz-Helmert functions Q2(Ψ),Q3(r),⋯,Q6(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS2(Ψ) − (sin Ψ/2)−1,S3(r) − (sinr/2R0)−1,⋯,S6(r) − 2R0/r andV2(Ψ) + (cos Ψ/2)/2($ sin^{2} $ Ψ/2),V3(r) + (cosr/2R0)/2($ sin^{2} $r/2R0),⋯,$$V_6 (r) + {{(R_0 \sqrt {4R_0^2 - r^2 } )} \mathord{\left/ {\vphantom {{(R_0 \sqrt {4R_0^2 - r^2 } )} {r^2 }}} \right. \kern-\nulldelimiterspace} {r^2 }}$$ illustrate the systematic errors in the”flat” Stokes function 2/Ψ or ”flat”Vening-Meinesz function −2/$ Ψ^{2} $. The newly derivedStokes functions S3(r),⋯,S6(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV3(r),⋯,V6(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type ”equidistant”, ”conformal” and ”equiareal”.