Complex-image integral

[amaurigmartins]

Concept note

1. Project the Sommerfeld kernels into a sum of exponentials
2. Use Lipschitz identity to circumvent the improper integral
3. Apply a convolution with a rect() to integrate over finite length
4. ???
5. Stonks

Links:

• https://mathworld.wolfram.com/LipschitzsIntegral.html
• https://github.com/andreasvarga/MatrixPencils.jl

[amaurigmartins]

Rallis DCIM approximation of the mixed Pollaczek integral

Identification and source

Field	Value
Family	External impedance
Geometry	Filaments at overhead height h_1, burial depth h_2 and horizontal spacing x.
Calculated quantities	Finite rational complex-image sum for mutual earth-return impedance between one overhead and one buried conductor
Earth structure	Homogeneous conductive half-space below air.
Model and approximation	GPOF fits the spectral factor over a finite sampling interval and (4.14), \int_0^\infty e^{-p\lambda}\cos(q\lambda)d\lambda=p/(p^2+q^2), integrates each term. The two-level sampling variant is an empirical refinement.
Main source	K. V. Rallis (2012) for the DCIM/GPOF representation; Pollaczek for the parent kernel
Citation key(s)	:Rallis2013
Evidence status	Original-thesis page image verified

Description. DCIM/GPOF approximation of the mixed overhead/buried Pollaczek integral, yielding a short sum of rational complex-image terms.

Assumptions.

Field	Treatment	Evidence
Impressed longitudinal propagation constant Γ	Neglected by the Pollaczek parent.	Stated through (4.4).
Air propagation constant γ_air	Not retained.	Stated by parent reduction.
Earth propagation constant γ_earth	k^2=j\omega\mu_0\sigma.	Stated — §4.2.
Earth permittivity and displacement current	Neglected.	Stated — definition of k.
Range of validity	Demonstrated with 100 samples, 14 terms and ``T_0=28	k
Earth permeability μ_earth	\mu_0.	Stated — (4.3).
Arrangement	Mutual term between one overhead and one underground parallel conductor.	Stated — Fig. 4.2.
Earth structure	Homogeneous conductive half-space below air.	Stated — Fig. 4.2.
Conductor and insulation geometry	Filaments at overhead height h_1, burial depth h_2 and horizontal spacing x.	Stated — Fig. 4.2.
Constitutive and field assumptions	Linear isotropic soil; quasi-static classical kernel; finite sum is an evaluator approximation.	Stated/inherited.
Conventions	j=\sqrt{-1}, positive height/depth, per-unit-length mutual impedance.	Stated — (4.3)–(4.4).

Expression. The mixed kernel is approximated by

$$\frac{e^{-h_2\sqrt{\lambda^2+k^2}}} {\lambda+\sqrt{\lambda^2+k^2}} \approx\sum_{n=1}^{N}c_ne^{s_n\lambda}, \qquad\text{(4.11)}$$ $$\begin{aligned} J_{ua}&\approx2\sum_{n=1}^{N}c_n\frac{H_n}{H_n^2+x^2} \\\ H_n&=h_1-s_n \\\ n&=1,2,\ldots,N, \end{aligned}\qquad\text{(4.15)}$$

and Z_{ua}=j\omega\mu_0J_{ua}/(2\pi).

Approximation. GPOF fits the spectral factor over a finite sampling interval and (4.14), \int_0^\infty e^{-p\lambda}\cos(q\lambda)d\lambda=p/(p^2+q^2), integrates each term. The two-level sampling variant is an empirical refinement.

Limitations. Mutual mixed geometry only. Coefficients must be regenerated from the fit; no error theorem or universal coefficient set is supplied. Displacement current, finite radii, layering and longitudinal propagation are absent.

Reference. Rallis2013. K. V. Rallis, doctoral thesis, 2012, DOI 10.12681/eadd/34633, equations (4.3)–(4.4), (4.10)–(4.15), printed pp. 69, 75–76.

Transcription source. Original thesis page images. The fitted numerator, choice of \lambda rather than \sqrt{\lambda^2+k^2}, rational denominator, factor two and H_n=h_1-s_n were visually verified.

Source transcription

$$J_{ua}=2\int_0^\infty \frac{e^{-h_1\lambda-h_2\sqrt{\lambda^2+k^2}}} {\lambda+\sqrt{\lambda^2+k^2}}\cos(x\lambda)\,d\lambda. \qquad\text{(4.10)}$$

Notation map

Source symbol	Display symbol	Physical meaning	Units/convention
J_{ua}	unchanged	mixed Pollaczek correction	dimensionless
h_1,h_2,x	unchanged	overhead height, burial depth, horizontal spacing	m
c_n,s_n	unchanged	fitted residues and complex poles	fit-dependent
H_n	unchanged	effective complex overhead height	m

No notation was renamed.

Numerical interpretation

Select EarthImpedance.Formula(:Pollaczek1926;evaluation=:dcim) for
one-level GPOF, or evaluation=:dcim_two_level for two-level fitting.
These selections change all three placement evaluators, not the physical
Pollaczek assumptions. The default spectral evaluator is unchanged.

The fit uses 100 uniformly spaced samples and at most 14 retained singular
directions per level. Appendix A defines the shifted sample matrices
Y_1,Y_2; for Y_1=UDV^H, the image poles follow from the
eigenvalues of D^{-1}U^HY_2V. A least-squares solve determines the
residues. Numerically unresolved directions and growing exponentials are
excluded before the residue solve. dcim_samples and dcim_terms
expose the sample and rank limits.

The sampling variable is divided by |k| before fitting. The
two-level implementation first fits the far interval, then fits its
residual on the near interval. Each retained residue is stored with its
sampling origin; this avoids forming overflowing coefficients before
multiplication by a decaying Bessel function. Coefficients are generated
in double precision, including for higher-precision output arithmetic.
Neither higher output precision nor the sample count constitutes an
error bound for the finite fit.

The fitted spectral function is integrated to infinity as prescribed by
the image sum. No parent-integral fallback is hidden behind this selection.
Tests separately verify the matrix-pencil recovery, integration of the
fitted function, parent comparisons, and matrix assembly. In particular,
large lateral separation can expose substantial buried-conductor fit
error; the spectral or exact-series selections remain available.

The one-level interval is 0\le\lambda/|k|\le28; the two-level
intervals are [5,100] and [0,5]. Burial depth remains
inside the fitted spectral numerator, so mixed coefficients are
regenerated for that depth. Exchanging source and receiver preserves
the same height and depth assignments.

Evidence and approximation sources

Rallis contributes the DCIM fit and finite sum; the underlying kernel remains Pollaczek's. This record is an approximation formula, not a priority claim for the mixed physical model.

Limitations and discrepancies

• The source's empirical GPOF settings are not validity guarantees.

[amaurigmartins]

Rallis DCIM approximation of the underground Pollaczek integral

Identification and source

Field	Value
Family	External impedance
Geometry	Filament parent; depths h_1,h_2, horizontal separation x.
Calculated quantities	Finite discrete-complex-image sum for the homogeneous-earth buried/buried Pollaczek correction
Earth structure	Homogeneous conductive half-space below air.
Model and approximation	Complex poles s_n and residues c_n are fitted with GPOF after mapping \gamma=P_at+P_b over t\in[0,1]; (4.8) then integrates each exponential analytically. This is a data-fitted finite-sum approximation, not an identity.
Main source	K. V. Rallis (2012) for the DCIM/GPOF representation; Pollaczek for the physical kernel
Citation key(s)	:Rallis2013
Evidence status	Original-thesis page image verified

Description. Discrete Complex Image Method approximation that fits the Pollaczek spectral denominator by exponentials and integrates each resulting buried/buried term analytically into a finite K_1 sum.

Assumptions.

Field	Treatment	Evidence
Impressed longitudinal propagation constant Γ	Neglected by the adopted Pollaczek kernel.	Stated through (4.1)–(4.2).
Air propagation constant γ_air	Not retained.	Stated by the conduction-only parent.
Earth propagation constant γ_earth	k^2=j\omega\mu_0\sigma.	Stated below (4.1), p. 68.
Earth permittivity and displacement current	Neglected.	Stated by k^2=j\omega\mu_0\sigma.
Range of validity	Thesis uses 100 samples, endpoint ``T_0=10	k
Earth permeability μ_earth	\mu_0.	Stated — (4.1).
Arrangement	Underground; self/mutual parallel conductors.	Stated — Fig. 4.1.
Earth structure	Homogeneous conductive half-space below air.	Stated — Fig. 4.1 and (4.2).
Conductor and insulation geometry	Filament parent; depths h_1,h_2, horizontal separation x.	Stated — Fig. 4.1.
Constitutive and field assumptions	Linear isotropic soil, classical quasi-static Pollaczek assumptions; DCIM is a numerical approximation.	Stated/inherited — §4.2.
Conventions	\gamma=\sqrt{\lambda^2+k^2}; positive burial depths; per-unit-length impedance.	Stated — (4.2), (4.5).

Expression. The fitted denominator and resulting finite sum are

$$\frac{1}{\lambda+\sqrt{\lambda^2+k^2}} \approx\sum_{n=1}^{N}c_ne^{s_n\sqrt{\lambda^2+k^2}}, \qquad\text{(4.5)}$$ $$\begin{aligned} J_{uu}&\approx\sum_{n=1}^{N} \frac{2c_nkH_n}{\sqrt{H_n^2+x^2}} K_1\!\left(k\sqrt{H_n^2+x^2}\right) \\\ H_n&=h_1+h_2-s_n. \end{aligned}\qquad\text{(4.9)}$$

The source inserts this in Z_{uu}=j\omega\mu_0[K_0(kd)-K_0(kD)+J_{uu}]/(2\pi).

Approximation. Complex poles s_n and residues c_n are fitted with GPOF after mapping \gamma=P_at+P_b over t\in[0,1]; (4.8) then integrates each exponential analytically. This is a data-fitted finite-sum approximation, not an identity.

Limitations. Coefficients are generated numerically and not tabulated as universal constants. Accuracy depends on sample path, term count, frequency and separation. The inherited physics is filamentary, homogeneous-earth and conduction-only.

Reference. Rallis2013. K. V. Rallis, doctoral thesis, 2012, DOI 10.12681/eadd/34633, equations (4.1)–(4.9), printed pp. 68–70.

Transcription source. Original thesis page images. The fitted variable, exponent sign as printed, K_1 argument, prefactor and complex-distance definition were visually verified.

Source transcription

The identity used termwise is

$$\int_0^\infty e^{-\beta\sqrt{\gamma^2+x^2}}\cos(bx)\,dx =\frac{\beta\gamma}{\sqrt{\beta^2+b^2}} K_1\!\left(\gamma\sqrt{\beta^2+b^2}\right). \qquad\text{(4.8)}$$

Notation map

Source symbol	Display symbol	Physical meaning	Units/convention
J_{uu}	unchanged	buried/buried Pollaczek interface integral	dimensionless
k	unchanged	conduction-only earth constant	\mathrm m^{-1}
c_n,s_n	unchanged	GPOF residues and poles	fit-dependent
H_n	unchanged	complex image depth	m

No notation was renamed.

Numerical interpretation

Select EarthImpedance.Formula(:Pollaczek1926;evaluation=:dcim) for
one-level GPOF, or evaluation=:dcim_two_level for two-level fitting.
These selections change all three placement evaluators, not the physical
Pollaczek assumptions. The default spectral evaluator is unchanged.

The fit uses 100 uniformly spaced samples and at most 14 retained singular
directions per level. Appendix A defines the shifted sample matrices
Y_1,Y_2; for Y_1=UDV^H, the image poles follow from the
eigenvalues of D^{-1}U^HY_2V. A least-squares solve determines the
residues. Numerically unresolved directions and growing exponentials are
excluded before the residue solve. dcim_samples and dcim_terms
expose the sample and rank limits.

The sampling variable is divided by |k| before fitting. The
two-level implementation first fits the far interval, then fits its
residual on the near interval. Each retained residue is stored with its
sampling origin; this avoids forming overflowing coefficients before
multiplication by a decaying Bessel function. Coefficients are generated
in double precision, including for higher-precision output arithmetic.
Neither higher output precision nor the sample count constitutes an
error bound for the finite fit.

The fitted spectral function is integrated to infinity as prescribed by
the image sum. No parent-integral fallback is hidden behind this selection.
Tests separately verify the matrix-pencil recovery, integration of the
fitted function, parent comparisons, and matrix assembly. In particular,
large lateral separation can expose substantial buried-conductor fit
error; the spectral or exact-series selections remain available.

The one-level path is
\gamma/|k|=\kappa+t(\sqrt{100+\kappa^2}-\kappa),
where \kappa=k/|k|. Two-level endpoints correspond to
T_{02}/|k|=0.22 and T_{01}/|k|=10.
The affine spans are the differences between successive endpoints:
\sqrt{100+\kappa^2}-\sqrt{0.22^2+\kappa^2} and
\sqrt{0.22^2+\kappa^2}-\kappa.
These differences implement the connected paths in Fig. 4.4. The
unnumbered printed spans on p. 72 use a plus sign in the first path
and omit the subtraction of k in the second; they would not
connect the stated endpoints.

For H_n=H-s_n, the branch of
\sqrt{H_n^2+x^2} is continuous from positive H_n.
The shifted residue and scaled K_1 are multiplied before their
exponentials are expanded. The original K_0(kd)-K_0(kD)
term is unchanged.

Evidence and approximation sources

The thesis explicitly presents DCIM as its method; Pollaczek supplies the physical integral and Gradshteyn–Ryzhik the termwise identity. The record therefore classifies (4.9) as a new representation/evaluator, not new ground-return physics.

[amaurigmartins]

Rallis DCIM approximation of Carson's integral

Identification and source

Field	Value
Family	External impedance
Geometry	Filament parent at heights h_1,h_2 and horizontal separation x.
Calculated quantities	Finite rational complex-image sum for the homogeneous-earth overhead Carson correction
Earth structure	Homogeneous conductive half-space.
Model and approximation	The denominator is fitted by complex exponentials with GPOF and each is integrated using the elementary Laplace–cosine identity. The exact Struve/Bessel form (4.18), separately reproduced by the thesis, is not attributed to this DCIM method and is already represented elsewhere in the corpus.
Main source	K. V. Rallis (2012) for the DCIM/GPOF evaluator; Carson for the parent kernel
Citation key(s)	:Rallis2013
Evidence status	Original-thesis page image verified

Description. DCIM/GPOF finite-sum approximation to Carson's overhead mutual correction, retained separately from the exact Struve/Bessel identity that the thesis also reproduces.

Assumptions.

Field	Treatment	Evidence
Impressed longitudinal propagation constant Γ	Neglected by Carson's parent.	Stated through (4.17).
Air propagation constant γ_air	Not retained.	Stated by parent reduction.
Earth propagation constant γ_earth	k^2=j\omega\mu_0\sigma.	Stated below (4.17).
Earth permittivity and displacement current	Neglected.	Stated — definition of k.
Range of validity	Fit uses a finite, empirically chosen sampling interval and term count; no global error bound is provided.	Stated — §4.4.
Earth permeability μ_earth	\mu_0.	Stated — (4.16).
Arrangement	Overhead; mutual parallel conductors.	Stated — Fig. 4.7.
Earth structure	Homogeneous conductive half-space.	Stated — §4.4.
Conductor and insulation geometry	Filament parent at heights h_1,h_2 and horizontal separation x.	Stated — Fig. 4.7.
Constitutive and field assumptions	Linear isotropic conduction-only soil; quasi-static Carson kernel.	Stated/inherited.
Conventions	H=h_1+h_2; per-unit-length impedance.	Stated — (4.18).

Expression. The DCIM fit and final Carson correction are

$$\frac{1}{\lambda+\sqrt{\lambda^2+k^2}} \approx\sum_{n=1}^{N}c_ne^{s_n\lambda}, \qquad\text{(4.20)}$$ $$\begin{aligned} J_c&\approx2\sum_{n=1}^{N}c_n\frac{H_n}{H_n^2+x^2} \\\ H_n&=h_1+h_2-s_n \\\ n&=1,2,\ldots,N, \end{aligned}\qquad\text{(4.22)}$$

with Z=j\omega\mu_0[\ln(D/d)+J_c]/(2\pi).

Approximation. The denominator is fitted by complex exponentials with GPOF and each is integrated using the elementary Laplace–cosine identity. The exact Struve/Bessel form (4.18), separately reproduced by the thesis, is not attributed to this DCIM method and is already represented elsewhere in the corpus.

Limitations. Mutual overhead geometry is shown. Fit coefficients are numerical and setup-dependent. This representation inherits Carson's homogeneous, filamentary, conduction-only and quasi-static restrictions.

Reference. Rallis2013. K. V. Rallis, doctoral thesis, 2012, DOI 10.12681/eadd/34633, equations (4.16)–(4.22), printed pp. 78–80.

Transcription source. Original thesis page images. The fit, factor two, rational complex-image term and H_n=h_1+h_2-s_n were visually verified.

Source transcription

$$J_c=\int_0^\infty \frac{2e^{-(h_1+h_2)\lambda}} {\lambda+\sqrt{\lambda^2+k^2}}\cos(x\lambda)\,d\lambda. \qquad\text{(4.17)}$$

Notation map

Source symbol	Display symbol	Physical meaning	Units/convention
J_c	unchanged	Carson correction	dimensionless
d,D	unchanged	direct and image distances	m
c_n,s_n	unchanged	fitted residues and poles	fit-dependent
H_n	unchanged	complex image height	m

No notation was renamed.

Numerical interpretation

Select EarthImpedance.Formula(:Pollaczek1926;evaluation=:dcim) for
one-level GPOF, or evaluation=:dcim_two_level for two-level fitting.
These selections change all three placement evaluators, not the physical
Pollaczek assumptions. The default spectral evaluator is unchanged.

The fit uses 100 uniformly spaced samples and at most 14 retained singular
directions per level. Appendix A defines the shifted sample matrices
Y_1,Y_2; for Y_1=UDV^H, the image poles follow from the
eigenvalues of D^{-1}U^HY_2V. A least-squares solve determines the
residues. Numerically unresolved directions and growing exponentials are
excluded before the residue solve. dcim_samples and dcim_terms
expose the sample and rank limits.

The sampling variable is divided by |k| before fitting. The
two-level implementation first fits the far interval, then fits its
residual on the near interval. Each retained residue is stored with its
sampling origin; this avoids forming overflowing coefficients before
multiplication by a decaying Bessel function. Coefficients are generated
in double precision, including for higher-precision output arithmetic.
Neither higher output precision nor the sample count constitutes an
error bound for the finite fit.

The fitted spectral function is integrated to infinity as prescribed by
the image sum. No parent-integral fallback is hidden behind this selection.
Tests separately verify the matrix-pencil recovery, integration of the
fitted function, parent comparisons, and matrix assembly. In particular,
large lateral separation can expose substantial buried-conductor fit
error; the spectral or exact-series selections remain available.

The one-level interval is 0\le\lambda/|k|\le28, following
the construction of §4.3.2 used by §4.4. Two-level intervals are
[5,100] and [0,5]. The conductor radius enters only
the perfect-ground self logarithm; the self correction sets x=0.

Evidence and approximation sources

The exact special-function equation (4.18) is a previously known evaluation; (4.20)–(4.22) are the thesis's DCIM representation. They are kept distinct to avoid claiming new physics or duplicating the exact identity.

Limitations and discrepancies

• The thesis heading says DCIM even though an exact special-function evaluation is also available; the record captures only the fitted DCIM formula.

[amaurigmartins]

mop.1209.pdf