### Relation to incomplete cylindrical functions

Here we show the relationship of the Chapman function to several families of the incomplete cylindrical functions. To start with, we change the integration variable *t* in Eq. (1) according to \(s=\csc (t)\). In this way we arrive at the alternative integral representation of the Chapman function

$$\begin{aligned} {\text {Ch}}(X,\zeta ) = X\sin \zeta e^X\int \limits _{\cosh \beta }^\infty \exp \left( - X s\sin \zeta \right) s(s^2-1)^{-1/2}\,{\text {d}}s, \end{aligned}$$

(6)

where

$$\begin{aligned} \beta = \ln \frac{1+\cos \zeta }{\sin \zeta } =\ln \cot \frac{\zeta }{2} =\cosh ^{-1}(\csc \zeta ). \end{aligned}$$

(7)

Integrating Eq. (6) by parts we obtain

$$\begin{aligned} {\text {Ch}}(X,\zeta )=-X\sin \zeta \sinh \beta +X^2\sin ^2\zeta e^X\int \limits _{\cosh \beta }^\infty \exp \left( -s X \sin \zeta \right) \sqrt{s^2-1}\, {\text {d}}s. \end{aligned}$$

(8)

From this formula it is easy to derive the result (5) for \(\zeta =\pi /2\), i.e., \(\cosh \beta =1\), by using the integral representation of the Macdonald function (Abramowitz and Stegun 1972, Eq. 9.6.23)

$$\begin{aligned} K_n(x)=\frac{\sqrt{\pi }\left( \frac{1}{2} x\right) ^n}{\Gamma \left( n+\frac{1}{2}\right) }\int \limits _1^\infty \exp \left( - sx\right) \left( s^2-1\right) ^{n-\frac{1}{2}}\,{\text {d}}s, \end{aligned}$$

(9)

where \(\Gamma (\nu )\) is the Gamma function.

In the view of Eq. (9) the Chapman function (8) can be rewritten as

$$\begin{aligned} {\text {Ch}}(X,\zeta )=-X\sin \zeta \sinh \beta +X\sin \zeta e^X\left[ K_1(X\sin \zeta )-L_1\left( X\sin \zeta , \cosh \beta \right) \right] \end{aligned}$$

(10)

with

$$\begin{aligned} L_n(x, y)=\frac{\sqrt{\pi }\left( \frac{1}{2} x\right) ^n}{\Gamma \left( n+\frac{1}{2}\right) }\int \limits _1^y \exp \left( - s x\right) \left( s^2-1\right) ^{n-\frac{1}{2}} \,{\text {d}}s \end{aligned}$$

(11)

being the incomplete Macdonald function in the form introduced in Agrest and Maximov (1971). The nomenclature “incomplete cylindrical function” or “incomplete Bessel function” becomes clear from this definition as only in the limit \(y\rightarrow \infty\) the function (11) becomes the “complete” Macdonald function (9).

In order to relate the Chapman function to other types of incomplete cylindrical functions we perform the variable change in (8) according to \(s=\cosh \theta\) and obtain

$$\begin{aligned} {\text {Ch}}(X,\zeta )=-X\sin \zeta \sinh \beta +X^2\sin ^2\zeta e^X\int \limits _{\beta }^\infty \exp \left( - X \sin \zeta \cosh \theta \right) \sinh ^2\theta \,{\text {d}}\theta . \end{aligned}$$

(12)

This expression allows one to express the Chapman function through the incomplete Macdonald functions

$$\begin{aligned} K_n(x,y)=\int \limits _y^\infty e^{-x \cosh \theta }\cosh (n \theta )\,{\text {d}}\theta \end{aligned}$$

(13)

as defined by Jones (2007) or

$$\begin{aligned} \tilde{K}_n(x, y) = \int \limits _1^\infty e^{-(x t+ y/t)}t^{n-1}\,{\text {d}}t \end{aligned}$$

(14)

as defined by Terras (1981). Namely, using the definition for \(K_n(x,y)\) the Chapman function can be represented as

$$\begin{aligned} {\text {Ch}}(X,\zeta )=-X\sin \zeta \sinh \beta +\frac{1}{2}X^2\sin ^2\zeta e^X\Bigl [K_2(X\sin \zeta ,\beta )-K_0(X\sin \zeta ,\beta )\Bigr ]. \end{aligned}$$

(15)

The corresponding relationship between the Chapman function and the incomplete cylindrical function (14) is easily obtained from (15) by observing the obvious relation

$$\begin{aligned} K_n(x, y) = \frac{1}{2}\left[ e^{n y} \tilde{K}_n\left( \frac{x}{2} e^y, \frac{x}{2} e^{-y}\right) + e^{-n y} \tilde{K}_{-n}\left( \frac{x}{2} e^y, \frac{x}{2} e^{-y}\right) \right] . \end{aligned}$$

(16)

The recurrence relations also allow to establish the relationship between the incomplete Bessel functions in form of Agrest and Maximov, Jones, and Terras. For example, we have in our case of interest the following relations

$$\begin{aligned} L_0(x,\cosh y) = K_0(x)-K_0(x,y), \end{aligned}$$

(17)

and

$$\begin{aligned} L_1(x,\cosh y) = K_1(x)-K_1(x,y)+\tanh y\frac{\partial }{\partial y}K_1(x,y). \end{aligned}$$

(18)

We also note that the limiting case (5) for \(\zeta =\pi /2\) is obtained from (15) by observing that \(K_n(x,0)=K_n(x)\)^{Footnote 1} and with the use of the well-known recurrence relations for the Bessel functions.

The recently discovered relationship of the Chapman function to the solution of an inhomogeneous modified Bessel’s equation (Huestis 2001) yields the following representation of this mapping function:

$$\begin{aligned} {\text {Ch}}(X,\zeta ) = X\sin \zeta \cos \zeta e^X\int \limits _X^\infty e^{-t}\Bigl [I_1(X\sin \zeta )K_0(t\sin \zeta )+K_1(X\sin \zeta )I_0(t\sin \zeta )\Bigr ]\,{\text {d}}t, \end{aligned}$$

(19)

where \(I_n(x)\) are the modified Bessel functions of *n*th order. The advantage of this integral representation is that the Champan function in this formulation is stable for all values of *X* and \(\zeta\) as has been shown by Huestis (2001). The integrals in Eq. (19)

$$\begin{aligned} \mathcal {I}_1(X,\zeta ) = \int \limits _X^\infty e^{-t}I_0(t\sin \zeta )\,{\text {d}}t,\quad \mathcal {I}_2(X, \zeta )=\int \limits _X^\infty e^{-t}K_0(t\sin \zeta )\,{\text {d}}t \end{aligned}$$

(20)

cannot be analytically integrated in general case except for \(X=0\). One can show that these integrals possess a weak singularity for \(t\rightarrow \infty , \zeta =\pi /2\) while \(\mathcal {I}_2\) possesses an integrable singularity for \(t\rightarrow 0, \zeta \rightarrow 0\).

Equation (19) allows one to establish the relation of the Chapman function and another class of the incomplete cylindrical functions known as the incomplete Lipschitz–Hankel integrals (Agrest and Maximov 1971). Namely, the integrals (20) can be expressed through the incomplete Lipschitz–Hankel integrals (Agrest and Maximov 1971)

$$\begin{aligned} Ie_0(x, y) = \int \limits _0^y e^{- x t}I_0(t)\,{\text {d}}t ,\quad Ke_0(x, y) = \int \limits _0^y e^{- x t}K_0(t)\,{\text {d}}t \end{aligned}$$

(21)

as

$$\begin{aligned} \mathcal {I}_1(X,\zeta )= & {} \sec \zeta -\csc \zeta \, Ie_0\left( \csc \zeta , X\sin \zeta \right) ,\quad 0\le \zeta <\pi /2, \end{aligned}$$

(22)

$$\begin{aligned} \mathcal {I}_2(X,\zeta )= & {} \beta \sec \zeta -\csc \zeta \, Ke_0\left( \csc \zeta , X\sin \zeta \right) ,\quad 0\le \zeta <\pi /2. \end{aligned}$$

(23)

Here the parameter \(\beta\) is given by Eq. (7). Substituting Eqs. (22), (23) in Eq. (19) we obtain

$$\begin{aligned} {\text {Ch}}(X,\zeta )= & {} X\sin \zeta e^X\Biggl \{I_1(X\sin \zeta )\beta +K_1(X\sin \zeta )\Biggr .\nonumber \\&-\Bigl .\cot \zeta \Bigl [I_1(X\sin \zeta )Ke_0(\csc \zeta ,X\sin \zeta )+K_1(X\sin \zeta )Ie_0(\csc \zeta ,X\sin \zeta )\Bigr ]\Biggr \}. \end{aligned}$$

(24)

This representation reduces \({\text {Ch}}(X,\pi /2)\) directly to the limiting case (5).

Using Eq. (19) we now derive the relationship of the Chapman function to the incomplete Bessel functions (13). We start with the result obtained in Ref. Agrest and Maximov (1971):

$$\begin{aligned} L_0(x,\cosh y) = y I_0(x)+\sinh y\int \limits _0^x \left[ K_0(x)I_0(t)e^{-t\cosh y}-I_0(x)K_0(t)e^{-t\cosh y}\right] \,{\text {d}}t, \end{aligned}$$

(25)

where the function \(L_0\) is defined in Eq. (11). Differentiating both sides of Eq. (25) with respect to *x* and using the relations \({\text {d}}I_0(x)/{\text {d}}x=I_1(x)\) and \({\text {d}}K_0(x)/{\text {d}}x=-K_1(x)\) we obtain

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}x}L_0(x, \cosh y) = y I_1(x) -\sinh y\int \limits _0^x \left[ K_1(x)I_0(t)e^{-t\cosh y}+I_1(x)K_0(t)e^{-t\cosh y}\right] \,{\text {d}}t. \end{aligned}$$

(26)

This equation allows one to rewrite the Chapman function (19) as

$$\begin{aligned} {\text {Ch}}(X,\zeta ) = \frac{1}{2}X\sin \zeta e^X\left[ K_1(X\sin \zeta ,\beta )+K_{-1}(X\sin \zeta ,\beta )\right] =X\sin \zeta e^X K_1(X\sin \zeta ,\beta ), \end{aligned}$$

(27)

where the property \(K_{-n}(x,y)=K_n(x,y)\) has been used. This is alternative to Eq. (15) representation of the Chapman function in terms of the incomplete Macdonald functions (13).

To conclude this section we relate the Chapman function with another class of the incomplete cylindrical functions, namely the incomplete Weber integrals (Agrest and Maximov 1971)

$$\begin{aligned} \widetilde{Q}_n(x,y)=(2x)^{-n-1}e^{x}\int \limits _0^y t^{n+1} I_n(t)\exp \left( -\frac{t^2}{4x}\right) \, {\text {d}}t. \end{aligned}$$

(28)

These integrals appear in the theories of optical diffraction, pulsed radar theories, and mathematical statistics, see e.g. Vasylyev et al. (2013). Reference (Agrest and Maximov 1971) derives the relationship between the incomplete Weber integral \(\widetilde{Q}_0(x,y)\) and the incomplete Lipschitz–Hankel integral \(Ie_{0}(x,y)\) as given in Eq. (21). In our notations this relationship for \(0\le \zeta \le \pi /2\) reads as:

$$\begin{aligned} Ie_{0}(\csc \zeta , X\sin \zeta ) = \tan \zeta \left\{ 2e^{-2 X\sin ^2\frac{\zeta }{2}}\widetilde{Q}_0\left( X\sin ^2\frac{\zeta }{2}, X\sin \zeta \right) -\left[ 1-e^{-X}I_0(X\sin \zeta )\right] \right\} . \end{aligned}$$

(29)

The incomplete integral \(Ke_0(x, y)\) can be expressed in terms of \(Ie_0(x, y)\) as shown in Ref. Agrest and Maximov (1971) as

$$\begin{aligned} Ke_0(\csc \zeta , X\sin \zeta )= & {} \frac{K_0(X\sin \zeta )}{I_0(X\sin \zeta )}Ie_0(\csc \zeta , X\sin \zeta )\nonumber \\&+\tan \zeta \frac{1}{I_0(X\sin \zeta )}\Bigl [\beta I_0(X\sin \zeta )-L_0(X\sin \zeta ,\cosh \beta )\Bigr ], \end{aligned}$$

(30)

where \(L_n(x,y)\) is given by (11). Combining Eq. (29) with (30) and substituting them in (24) we obtain

$$\begin{aligned} {\text {Ch}}(X,\zeta )= & {} X\sin \zeta e^X\Biggl (\Biggr .K_1(X\sin \zeta )+\frac{I_1(X\sin \zeta )}{I_0(X\sin \zeta )} \left[ K_0(X\sin \zeta )-K_0(X\sin \zeta ,\beta )\right] \nonumber \\&-\left[ K_1(X\sin \zeta )-\frac{I_1(X\sin \zeta )}{I_0(X\sin \zeta )}K_0(X\sin \zeta )\right] \nonumber \\&\times \Biggl \{2e^{-2 X\sin ^2\frac{\zeta }{2}}\widetilde{Q}_0\left( X\sin ^2\frac{\zeta }{2}, X\sin \zeta \right) \Biggr .{-}\Biggl .\Biggl .\Bigl [1-e^{-X}I_0(X\sin \zeta )\Bigr ]\Biggr \}\Biggr ), \end{aligned}$$

(31)

where we have used Eq. (17) for the incomplete Bessel function \(L_0(x,y)\). The obtained expression is rather complex to be applied in practice. Its derivation serves merely for sake of completeness of the relations between the Chapman function and the incomplete cylindrical functions of different kinds. However, it appears that (31) becomes useful for the analytic approximation of the Chapman function when the scale parameter is small.

### Analytic approximations for the mapping function

In the previous section we have obtained several formulas that relate the Chapman mapping function (1) to the incomplete cylindrical functions of various types. Specifically, Eq. (10) relates the Chapman integral to the incomplete Macdonald function \(L_n(x,y)\) of the Agrest and Maksimov type, whereas Eqs. (15) and (27) express the former in terms of the incomplete Macdonald function \(K_n(x,y)\) of the Jones type or alternatively in terms of the Terras incomplete Macdonald function \(\tilde{K}_n(x, v)\). Finally, Eq. (24) is the representation in terms of the incomplete Lipschitz–Hankel integrals \(Ie_n(x,y), Ke_n(x,y)\) and can be also rewritten in terms of the incomplete Weber integral \(\widetilde{Q}_n(x,y)\) and the incomplete Macdonald function as given in Eq. (31). Since there are relationships among these various types of incomplete Bessel functions as discussed in the previous section, we consider the analytic approximation of the Chapman function using Eqs. (15) and (27) written in terms of \(K_n(x,y)\). The special case of small values of the scale parameter *X*, which is not of interest in atmospheric physics applications, is based on Eq. (31) and is considered in the Appendix A.

The choice of the incomplete Bessel function \(K_n(x,y)\) for the development of analytic approximations is dictated by suitability of its integral representation (13) for uniform asymptotic expansions. We refer to the work (Jones 2007) for details in derivation of the expansion formula and give here the lowest terms in the asymptotic expansion of \(K_n(x,y)\)

$$\begin{aligned} K_n(X\sin \zeta , \beta )\approx & {} \left( \frac{\pi }{2 X\sin \zeta }\right) ^{1/2}\Bigl (\alpha _0+\frac{\alpha _1}{X\sin \zeta }\Bigr )e^{-X\sin \zeta } {\text {erfc}}\left( \sqrt{2X\sin \zeta }\sinh \frac{\beta }{2}\right) \nonumber \\&+\left( \frac{\beta _0}{X\sin \zeta }+\frac{\beta _1}{X^2\sin ^2\zeta }\right) e^{-X}. \end{aligned}$$

(32)

Here \(\beta\) is given in Eq. (7) and

$$\begin{aligned} \alpha _0= & {} 1,\quad \alpha _1=\frac{1}{8}(4n^2-1), \end{aligned}$$

(33)

$$\begin{aligned} \beta _0= & {} \frac{\cosh n\beta }{\sinh \beta }-\frac{1}{2\sinh \frac{\beta }{2}},\quad \beta _1=\frac{n \sinh n\beta }{\sinh ^2\beta }-\frac{\cosh n\beta \cosh \beta }{\sinh ^3\beta }\nonumber \\&+\frac{1}{8\sinh ^3\frac{\beta }{2}}-\frac{4n^2-1}{16\sinh \frac{\beta }{2}}. \end{aligned}$$

(34)

This uniform asymptotic expansion of the incomplete Bessel function allows us to obtain the new analytic approximations of the Chapman integral, which are the main results of the present article. Substituting (32) in (15) yields

$$\begin{aligned} {\text {Ch}}(X,\zeta )\approx \sqrt{\frac{\pi }{2}}\sqrt{X\sin \zeta }e^{X(1-\sin \zeta )}{\text {erfc}}\left( \sqrt{2X\sin \zeta }\sinh \frac{\beta }{2}\right) + \coth \beta -\frac{1}{2\sinh \frac{\beta }{2}}. \end{aligned}$$

(35)

Inserting (32) in (27) we obtain

$$\begin{aligned} {\text {Ch}}(X,\zeta )\approx & {} \sqrt{\frac{\pi }{2}}\sqrt{X\sin \zeta }\,\Biggl (1+\frac{3}{8}\frac{1}{X\sin \zeta }\Biggr ) \,e^{X(1-\sin \zeta )}{\text {erfc}}\left( \sqrt{2X\sin \zeta }\sinh \frac{\beta }{2}\right) \nonumber \\&+ \coth \beta -\frac{1}{2\sinh \frac{\beta }{2}}-\frac{1}{X\sin \zeta } \Biggl (\frac{1}{\sinh ^3\beta }-\frac{1}{8\sinh ^3\frac{\beta }{2}}+\frac{3}{16\sinh \frac{\beta }{2}}\Biggr ),\nonumber \\ \beta= & {} \cosh ^{-1}(\csc \zeta ). \end{aligned}$$

(36)

We see that Eq. (32) is more suitable for deriving of asymptotic expansion of the Chapman function since it preserves the order of expansion of the incomplete Bessel function (27) in contrast to the definition (15).

Inspection of Eq. (36) shows that the first term on the r.h.s resembles the asymptotic expansion of the Macdonald function \(K_1(X\sin \zeta )\) with the complementary error function. This observation shows the similarity of the obtained asymptotic expansion formula (36) with the large-\(\zeta\) expansion of Chapman, cf. Eq.(38) in his original article (Chapman 1931). If one performs the expansion of the Macdonald function up to the second term in the first summand of the Chapman asymptotic expression, one obtains exactly the first term of Eq. (36). With this respect it is worth to note that Chapman considered the asymptotic of small and large zenith angles separately. Moreover, Chapman’s original asymptotic expansions are complex polynomial series with poor convergence, the issue corrected in Ref. Huestis (2001). Equation (36) originates from uniform asymptotic expansions and is valid for all values of zenith angle and is simpler to implement in practice.

To conclude this section we provide an explicit expression for approximation (36) in the form useful for numeric calculations

$$\begin{aligned} {\text {Ch}}(X,\zeta )\approx & {} \sqrt{\frac{\pi }{2}}\sqrt{X\sin \zeta }\,\Biggl (1+\frac{3}{8}\frac{1}{X\sin \zeta }\Biggr ) \,e^{X(1-\sin \zeta )}{\text {erfc}}\left[ \sqrt{X(1-\sin \zeta )}\right] \nonumber \\&+ \frac{1}{\sqrt{1-\sin ^2\zeta }}\Biggl \{ 1-t-\frac{1}{X\sin \zeta (1-\sin ^2\zeta )^2}\nonumber \\&\Biggl [\sin ^3\zeta -t^3+\frac{3}{8}t(1-\sin ^2\zeta )\Biggr ]\Biggr \},\nonumber \\ t= & {} \sqrt{\sin \zeta (1+\sin \zeta )/2}. \end{aligned}$$

(37)

Here the zenith angle enters as an argument of sine function only and the expression \(\sin \zeta\) is optimal to be evaluated prior the substitution in (37). Finally we note that for sufficiently large values of *X* as \(\zeta \rightarrow 0\) the formula (37) approaches the expected limit \(\sec \zeta\).