Bulirsch%0Aelliptic%20integrals
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1: 19.2 Definitions
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§19.2(i) General Elliptic Integrals
… ►§19.2(ii) Legendre’s Integrals
… ►§19.2(iii) Bulirsch’s Integrals
►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►§19.2(iv) A Related Function:
…2: 1.14 Integral Transforms
§1.14 Integral Transforms
… ►Sufficient conditions for the integral to converge are that is a positive real number, and as , where . … ►If the integral converges, then it converges uniformly in any compact domain in the complex -plane not containing any point of the interval . … ►If is absolutely integrable on for every finite , and the integral (1.14.47) converges, then … ►If is piecewise continuous on and the integral (1.14.47) converges, then …3: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
… ►Unless is a nonpositive integer, has a branch point at . For each branch of is an entire function of . … ►For and , … ►§8.19(x) Integrals
…4: 6.2 Definitions and Interrelations
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§6.2(i) Exponential and Logarithmic Integrals
… ►As in the case of the logarithm (§4.2(i)) there is a cut along the interval and the principal value is two-valued on . … ►In the next three equations . …( is undefined when , or when is not real.) … ►§6.2(ii) Sine and Cosine Integrals
…5: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions , , , and is a multivalued function of with branch point at . Furthermore, and are entire functions of , and and are meromorphic functions of with simple poles at and , respectively. … ►When (and when , in the case of , or , in the case of ) the principal values of , , , and are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►6: 7.2 Definitions
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§7.2(ii) Dawson’s Integral
… ►§7.2(iii) Fresnel Integrals
… ►Values at Infinity
… ►§7.2(iv) Auxiliary Functions
… ►§7.2(v) Goodwin–Staton Integral
…7: 19.16 Definitions
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§19.16(i) Symmetric Integrals
… ►where () is a real or complex constant, and …In (19.16.1)–(19.16.2_5), except that one or more of may be 0 when the corresponding integral converges. … ►with the same conditions on , , as for (19.16.1), but now . … ►When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an -function with one less variable: …8: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
►§7.18(i) Definition
… ►and for , … ► … ►Hermite Polynomials
…9: 36.2 Catastrophes and Canonical Integrals
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Canonical Integrals
… ►with the contour passing to the lower right of . …with the contour passing to the upper right of . … ► … ►§36.2(iii) Symmetries
…10: 19.39 Software
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