ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 9 pot

Section A.2 Evaluation of the Coefficients in the Potentials 85 (due to the fact that Cauchy’s Integral Theorem, which guarantees path inde- pendence in a simply-connected region, is not necessarily satisfied for contours around the hole). It follows that  L ∗ (z) dz = C c , (A.19) where C c is some complex constant (the same reasoning applies here as was used to show that the value B c in (A.13) is constant). Proceeding as before (see the reasoning used to determine equation (A.17)), the integral of ∗ (z) can be written as  z z 0 ∗ (z) dz = C c 2πi log(z − z c ) + ∗∗ ϕ(z), (A.20) where ∗∗ ϕ(z) is a single-valued analytic function. Substituting (A.20) in (A.18) and expanding gives ϕ(z) = A c z log(z − z c ) − A c z c log(z − z c ) − A c (z − z c ) + C c 2πi log(z − z c ) + ∗∗ ϕ(z)+ C. (A.21) Combining logarithmic terms and incorporating all single-valued analytic terms in a new analytic function, ∗ ϕ(z), results in ϕ(z) = A c z log(z − z c ) + γ c log(z − z c ) + ∗ ϕ(z), (A.22) where the constant γ c and the new single-valued analytic function ∗ ϕ(z) have been introduced for convenience. The multi-valued nature of the potential ψ(z) can be determined by noting that (z) = ψ  (z) is single-valued (this is because the stresses on the left-hand side of (2.5) and the function ϕ  (z) on the right-hand side of the same equation are all single valued). With the same reasoning used to determine (A.17) and (A.20), it can be shown that the integral of (z) can be written as ψ(z) =  z z 0 (z)dz = γ  c log(z − z c ) + ∗ ψ(z). (A.23) where γ  c is a complex constant and ∗ ψ(z) is a single-valued analytic function. §A.2 Evaluation of the Coefficients in the Potentials All that remains for a full determination of the multi-valued nature of the poten- tials in a region with a hole is to determine the unknown coefficients in (A.22) and (A.23). This is accomplished as follows. In order to ensure that the displacements associated with the multi-valued potentials given by (A.22) and (A.23) are single-valued, equation (2.1) must not 86 Multi-Valued Complex Potentials Appendix A become multi-valued anywhere in the region. Substituting (A.22) and (A.23) in (2.1) gives 2µ(u + iv) = κ  A c z log(z − z c ) + γ c log(z − z c ) + ∗ ϕ(z)  − z  A c log(z − z c ) + A c z z − z c + γ c z − z c + ∗ ϕ  (z)  − γ  c log(z − z c ) − ∗ ψ(z). (A.24) Separating the multi-valued parts from the single-valued parts results in 2µ(u + iv) = (κA c z + κγ c ) log(z − z c ) − (zA c + γ  c )log(z − z c ) + f(z), (A.25) where f(z) = κ ∗ ϕ(z) − A c zz z − z c − z γ c z − z c − z ∗ ϕ  (z) − ∗ ψ(z), (A.26) which is single-valued. Denoting by r zc the distance from z to z c , denoting by ϑ zc the argument of z − z c (see Figure A.3), and splitting the logarithms into their real and imaginary parts yields 2µ(u + iv) = (κzA c + κγ c )(ln r zc + iϑ zc ) − (zA c + γ  c )(ln r zc − iϑ zc ) + f(z). (A.27) After collecting terms, we have 2µ(u + iv) = (κzA c + κγ c − zA c − γ  c ) ln r zc + i(κzA c + κγ c + zA c + γ  c )ϑ zc + f(z). (A.28) The function ϑ zc in this expression increases continuously along each circuit around the point z c . The other functions in (A.28) are single-valued. It follows that the coefficient of ϑ zc must be set to zero in order to ensure single-valued displacements: A c (κ + 1)z + κγ c + γ  c = 0. (A.29) As a result A c = 0 (A.30) and κγ c + γ  c = 0. (A.31) The coefficients γ c and γ  c can be related to the resultant force on the hole through use of (2.6). We take an integration path consisting of a complete circuit around L h in a clockwise direction (see Figure A.3). Note that this integration path is chosen such that the region is to the left, as specified for (2.6). Denoting by [] L h the increase undergone by the expression inside the brackets along the Section A.2 Evaluation of the Coefficients in the Potentials 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y ϑ zc r zc z c L h Figure A.3: Integration path. integration path, integrating along L h in (2.6) and substituting equations (A.22) and (A.23) gives  γ c log(z − z c ) + ∗ ϕ(z) + z γ c z − z c + z ∗ ϕ  (z) + γ  c log(z − z c ) + ∗ ψ(z)  L h = i  L h (t x + it y ) ds = i( h F x + i h F y ), (A.32) where h F x + i h F y is the resultant force on the hole, and where we have utilized (A.30). Using the same notation as above, the logarithms can be split into their real and imaginary parts, resulting in  i(γ c − γ  c )ϑ zc + (γ c + γ  c ) ln r zc + g(z)  L h = i( h F x + i h F y ), (A.33) where g(z) = z γ c z − z c + ∗ ϕ(z) + z ∗ ϕ  (z) + ∗ ψ(z). (A.34) We note that ϑ zc decreases by an amount 2π upon making a complete circuit around L h , and that ln r zc and g(z) each return to their original values upon making a complete circuit around L h . The result is that (γ c − γ  c )(−2π) = h F x + i h F y . (A.35) Solving for γ c and γ  c using (A.31) and (A.35), applying (A.30), and substituting the results in (A.22) and (A.23) finally yields ϕ(z) =− h F x + i h F y 2π(1 + κ) log(z − z c ) + ∗ ϕ(z), (A.36) 88 Multi-Valued Complex Potentials Appendix A and ψ(z) = κ( h F x − i h F y ) 2π(1 + κ) log(z − z c ) + ∗ ψ(z). (A.37) §A.3 Extension to more than one Hole The potentials ϕ(z) and ψ(z) for a finite plane with multiple holes can be obtained by a superimposition of the potentials for a single hole (given by (A.36) and (A.36)). This is due to the fact that the coefficients of the logarithms (or more precisely, the integrals (A.13), (A.19), and (A.23) and the resultant force, given by the integral (A.32)) are not effected by the presence of other holes. It follows that the potentials for a finite plane with m holes can be written as ϕ(z) =− m  k=1 k F x + i k F y 2π(1 + κ) log(z − z k ) + ∗ ϕ n (z), (A.38) ψ(z) = m  k=1 κ( k F x − i k F y ) 2π(1 + κ) log(z − z k ) + ∗ ψ n (z), (A.39) where the values z k denote points within their respective holes. The functions ∗ ϕ n (z) and ∗ ψ n (z) denote new arbitrary single-valued analytic functions. Appendix B IMPLEMENTATION IN A COMPUTER PROGRAM The focus of this appendix is the implementation of the results from Chap- ter 4 in a computer program. In order to facilitate this implementation, the expressions from Chapter 4 are put into a form that is capable of generating dimensionless stresses and displacements. Using this form, we take the deriva- tives of the potentials and use them to write out the equations for the stresses and displacements explicitly. § B.1 Dimensionless Complex Potentials In order to generate dimensionless expressions for the stresses and the displace- ments, we divide the complex potentials by a constant P that has dimensions of force per unit length. This will generate dimensionless complex potentials on the basis of (2.1) – (2.5). The constant P will be defined according to the parameters appearing in the solution of particular problems. We start this process for the results derived in Chapter 4 by dividing the values of the coefficients in the Laurent expansions (4.27) and (4.28) by P .We make the following definitions: p k = a k P ,q k = b k P ,r k = c k P ,s k = d k P , (B.1) where a k and b k are the coefficients in the Laurent expansion (4.27), given by (4.49), (4.50), and (4.55). The coefficients c k and d k in the Laurent expan- sion (4.28) are given by (4.35) – (4.37). Dividing the Laurent expansions (4.27) and (4.28) by P results in ∗ ϕ 0 (ζ ) = p 0 + ∞  k=1 p k ζ k + ∞  k=1 q k ζ −k , (B.2) ∗ ψ 0 (ζ ) = r 0 + ∞  k=1 r k ζ k + ∞  k=1 s k ζ −k . (B.3) 89 90 Implementation in a Computer Program Appendix B where ∗ ϕ 0 (ζ ) = ϕ 0 m (ζ ) P , ∗ ψ 0 (ζ ) = ψ 0 m (ζ ) P . (B.4) Dividing (4.35) – (4.37) by P gives r 0 = B 0 ◦ P − p 0 − 1 2 p 1 − 1 2 q 1 , (B.5) r k = B −k ◦ P − q k − 1 2 (k + 1)p k+1 + 1 2 (k − 1)p k−1 ,k>0, (B.6) s k = B k ◦ P − p k + 1 2 (k − 1)q k−1 − 1 2 (k + 1)q k+1 ,k>0. (B.7) On the basis of (4.42) – (4.47) and (4.51), the only parameters in (4.49), (4.50), and (4.55) that are not dimensionless are k β i13 and k β i13 . These parameters have the dimensions of the Fourier coefficients B k ◦ and A k ◦ . Dividing equations (4.49) and (4.50) through by P results in p k = k β 011 p 0 + k β 012 p 0 + k−1  i=0 k β i13 P k = 1, 2, 3, , (B.8) q k = k β 021 p 0 + k β 022 p 0 + k−1  i=0 k β i23 P k = 1, 2, 3, (B.9) The expression for a 0 , equation (4.55), cannot be implemented in a computer program unless the limit k →∞is replaced by an expression like k = k I , where k I is some large integer, say 10,000. Dividing (4.55) by P and replacing the symbol ∞ with k I results in p 0 = k I β 012 k I −1  i=0 k I β i13 P − k I β 021 k I −1  i=0 k I β i23 P k I β 021 k I β 011 − k I β 022 k I β 012 . (B.10) We now have complete and dimensionless expressions for ∗ ϕ 0 (ζ ) and ∗ ψ 0 (ζ ) which can be programmed into a computer. Finally, we convert the complete potentials given by (4.56) and (4.57) to their dimensionless forms. This gives ∗ ϕ(ζ) =− κ( h∗ F x + i h∗ F y ) 2π(1 + κ) log(− i2a 1 − ζ ) − h∗ F x + i h∗ F y 2π(1 + κ) log(− i2aζ 1 − ζ ) + p 0 + ∞  k=1 p k ζ k + ∞  k=1 q k ζ −k , (B.11) Section B.2 Dimensionless Coordinates 91 and ∗ ψ(ζ) = h∗ F x − i h∗ F y 2π(1 + κ) log(− i2a 1 − ζ ) + κ( h∗ F x − i h∗ F y ) 2π(1 + κ) log(− i2aζ 1 − ζ ) + r 0 + ∞  k=1 r k ζ k + ∞  k=1 s k ζ −k , (B.12) where h∗ F x + i h∗ F y = h F x + i h F y P , ∗ ϕ(ζ) = ϕ m (ζ ) P , ∗ ψ(ζ) = ψ m (ζ ) P . (B.13) § B.2 Dimensionless Coordinates The coordinates in the z-plane can be written in dimensionless form by dividing (4.7) by h, the depth of the tunnel. This results in ∗ z = ∗ ω(ζ) =−i ∗ a 1 + ζ 1 − ζ , (B.14) where ∗ z = z/h and ∗ ω(ζ) are the dimensionless coordinates and dimensionless mapping function, respectively, and where we have defined the parameter ∗ a = 1 − α 2 1 + α 2 . (B.15) This parameter is simply the dimensionless form of (4.8). The dimensionless coordinates ∗ z can be obtained by using (B.14) instead of (4.7) to map the coordinates back from the ζ -plane. § B.3 Evaluation of the Stresses and Displacements It is most convenient to calculate (2.1) and (2.4) – (2.5) in the ζ -plane and then to relate the resulting values to the dimensionless coordinates ∗ z through use of (B.14). In order to perform these calculations we will need expressions for ∗ ϕ  (z), ∗ ϕ  (z), and ∗ ψ  (z) in terms of ζ . The derivative of ϕ(z) with respect to z in terms of ζ was derived in (4.14). Similar results can be obtained for ∗ ϕ  (z) and ∗ ψ  (z). They are ∗ ϕ  (z) = 1 h ∗ W 1 (ζ ) ∗ ϕ  (ζ ), ∗ ψ  (z) = 1 h ∗ W 1 (ζ ) ∗ ψ  (ζ ), (B.16) where we have defined the (dimensionless) function ∗ W 1 (ζ ) = h ω  (ζ ) = i (1 − ζ) 2 2 ∗ a . (B.17) 92 Implementation in a Computer Program Appendix B The chain rule can be used twice to determine ∗ ϕ  (z) in terms of ζ : ∗ ϕ  (ζ ) = d dζ  ∗ ϕ  (z)ω  (ζ )  = d dz ∗ ϕ  (z)· dz dζ ·ω  (ζ ) + ∗ ϕ  (z)ω  (ζ ). (B.18) It follows that ∗ ϕ  (z) = 1 h 2 [ ∗ W 1 (ζ )] 2 ∗ ϕ  (ζ ) − 1 h 2 ∗ W 3 (ζ ) ∗ ϕ  (ζ ), (B.19) where we have defined the (dimensionless) function ∗ W 3 (ζ ) = h 2 ω  (ζ ) [ω  (ζ )] 3 =− (1 − ζ) 3 2 ∗ a 2 . (B.20) The derivatives of (B.11) and (B.12) with respect to ζ are: ∗ ϕ  (ζ ) =− κ( h∗ F x + i h∗ F y ) 2π(1 + κ) · 1 1 − ζ − h∗ F x + i h∗ F y 2π(1 + κ) · 1 ζ(1 − ζ) + ∞  k=1 kp k ζ k−1 − ∞  k=1 kq k ζ −k−1 , (B.21) ∗ ψ  (ζ ) = h∗ F x − i h∗ F y 2π(1 + κ) · 1 1 − ζ + κ( h∗ F x − i h∗ F y ) 2π(1 + κ) · 1 ζ(1 − ζ) + ∞  k=1 kr k ζ k−1 − ∞  k=1 ks k ζ −k−1 , (B.22) and ∗ ϕ  (ζ ) =− κ( h∗ F x + i h∗ F y ) 2π(1 + κ) · 1 (1 − ζ) 2 + h∗ F x + i h∗ F y 2π(1 + κ) · 1 − 2ζ [ζ(1 − ζ)] 2 + ∞  k=1 k(k − 1)p k ζ k−2 + ∞  k=1 k(k + 1)q k ζ −k−2 . (B.23) Expressions for the displacements in terms of ζ can be obtained by dividing (2.1) by P and substituting (4.7), (B.16), (B.19), and (B.21) – (B.23) in the resulting expression. This gives 2µ P (u + iv) = κ ∗ ϕ(ζ) − ∗ ω(ζ) ∗ W 1 (ζ ) ∗ ϕ  (ζ ) − ∗ ψ(ζ). (B.24) Expressions for the stresses can be obtained by multiplying (2.4) and (2.5) by h/P and substituting the same expressions in the result. This yields h P (σ xx + σ yy ) = 2  ∗ W 1 (ζ ) ∗ ϕ  (ζ ) + ∗ W 1 (ζ ) ∗ ϕ  (ζ )  , (B.25) Section B.4 Comparison of Numerical and Analytical Solutions 93 and h P (σ yy − σ xx + 2iσ xy ) = 2  ∗ ω(ζ)[ ∗ W 1 (ζ )] 2 ∗ ϕ  (ζ ) − ∗ ω(ζ) ∗ W 3 (ζ ) ∗ ϕ  (ζ ) + ∗ W 1 (ζ ) ∗ ψ  (ζ )  . (B.26) The dimensionless stresses and displacements have been fully determined in terms of the mapping parameter ζ . The dimensionless coordinates x/h and y/h corresponding to the values calculated using (B.24) - (B.26) can be obtained by mapping ζ to ∗ z with (B.14). § B.4 Comparison of Numerical and Analytical Solutions A computer program implementing the equations in the preceding sections is approximately 40% slower than code which determines p 0 , p k and q k by solving the system of equations (4.38) - (4.39) numerically and examining the behavior of p k for large values of k and different values of p 0 , as performed by Ver- ruijt [42]. It should be noted that part of this difference is due to the fact that in Verruijt’s implementation it is assumed that all the coefficients are purely imaginary (so that all calculations involve only real variables) whereas the cal- culations in this implementation are fully complex (making it possible to include a non-vertical resultant force acting on the tunnel). It may be noted that both the numerical and the analytical computation of the coefficients of the Lau- rent series result in extremely small errors in the normalized displacements and stresses. Even so, there are some small differences between the two techniques, as can be seen in Table B.1, and as will be discussed below. Table B.1: Comparison of solutions for the ground loss problem with ν = 0.3 and with no resultant forces acting on the boundaries. The values have been normalized through use of the parameter P = 2µu g , where u g is the maximum convergence of the tunnel. Solution r/h Num. Terms Max. Error in u/u g Max. σ surface ·h/(2µu g ) analytic 0.01 4 9.9 × 10 −10 2.4 ×10 −15 numeric 0.01 4 1.5 ×10 −8 2.4 ×10 −15 analytic 0.26 11 3.3 ×10 −9 4.9 ×10 −15 numeric 0.26 10 3.4 ×10 −8 2.2 × 10 −13 analytic 0.9 41 1.5 × 10 −7 1.6 ×10 −13 numeric 0.9 38 3.4 × 10 −8 1.2 ×10 −12 94 Implementation in a Computer Program Appendix B Based on the ground loss problem with no resultant forces acting on the boundaries, the fully analytical solution is approximately one order of mag- nitude more accurate for common tunnel depths (r/h < 0.5) and about one order of magnitude less accurate for extremely shallow tunnels (r/h > 0.8). The surface stresses (σ yy and σ xy ) in the new solution are on the average of one order of magnitude more accurate for all values of r/h. As noted earlier, however, both solutions are very accurate and the differences discussed here are negligible in comparison to the small errors in the boundary conditions. [...]... the purpose of this appendix to derive the transformed Fourier expansion of the boundary conditions for the ovalization of a circular tunnel used in this thesis The transformation is made from the physical z-plane containing the tunnel to the conformally mapped annulus in the ζ -plane (see Figure C.1) using the mapping function (4.7) presented in Chapter 4 The results are written in the form of the given... Condition Along the Tunnel The displacements for the ovalization of a tunnel in the z-plane are given in terms of the polar coordinates r and θ (see the left side of Figure C.1) by ur + iuθ = uo cos 2θ, (C.1) where ur and uθ are the radial and tangential displacements along the tunnel boundary, and where uo is the maximum displacement of the tunnel boundary x ... Figure C.1: Ovalization of a tunnel in the z-plane; conformal mapping the ζ -plane 95 96 The Ovalization Boundary Condition Appendix C It is our goal to transform (7.1) to the ζ -plane, and then to expand it in a Fourier series in order to determine the function G (ασ ) given in (4.31) We start by writing (7.1) in terms of the horizontal and vertical displacements ut + ivt... boundary: h g(z) = ut + ivt on Lh (C.5) In (C.4) the given displacements along the tunnel have been written completely in terms of z and z, allowing substitution of the conformal mapping (4.7) for z in order to transform the given displacements to the ζ -plane § C.2 Transformation of the Boundary Conditions h The given displacements g(z) can now be transformed to the ζ -plane by substituting (4.7) for z... displacements g(z) can now be transformed to the ζ -plane by substituting (4.7) for z in (C.4) We use the following notation, introduced in Chapter 4: ζ = αeiϑ = ασ and ζ = ασ = ασ −1 , along the inner boundary of the annulus in the ζ -plane After substituting (4.7) for z in (C.4), combining fractions, expanding the numerator, and collecting terms we obtain −i c1 σ 3 + c2 σ 2 + c3 σ + c4 + c5 σ −1 + c6 σ −2 2α... + α 4 ), c2 = −α 4 (5 + 4α 2 + α 4 ), c3 = 4α 5 (4 + α 2 ), c4 = −4α 4 (1 + 4α 2 ), c5 = α 3 (1 + 4α 2 + 5α 4 ), c6 = −α 4 (1 + α 4 ), and where we have simplified the coefficient on the left-hand side of (C.6) by using the relation 1 + α2 α = , (C.7) 2h r . and ∗ ψ(z) is a single-valued analytic function. §A.2 Evaluation of the Coefficients in the Potentials All that remains for a full determination of the multi-valued nature of the poten- tials in a region. system of equations (4.38) - (4. 39) numerically and examining the behavior of p k for large values of k and different values of p 0 , as performed by Ver- ruijt [42]. It should be noted that part of. ×10 −15 analytic 0.26 11 3.3 ×10 9 4 .9 ×10 −15 numeric 0.26 10 3.4 ×10 −8 2.2 × 10 −13 analytic 0 .9 41 1.5 × 10 −7 1.6 ×10 −13 numeric 0 .9 38 3.4 × 10 −8 1.2 ×10 −12 94 Implementation in a Computer