# §19.3 Graphics

## §19.3(i) Real Variables

See Figures 19.3.119.3.6 for complete and incomplete Legendre’s elliptic integrals.

Figure 19.3.1: and as functions of for . Graphs of and are the mirror images in the vertical line .
Figure 19.3.2: and the Cauchy principal value of for . Both functions are asymptotic to as ; see (19.2.19) and (19.2.20). Note that , .
Figure 19.3.3: as a function of and for , . If (), then the function reduces to , becoming infinite when . If (), then it has the value : put in (19.25.5) and use (19.25.1).
Figure 19.3.4: as a function of and for , . If (), then the function reduces to , with value 1 at . If (), then it has the value , with limit 1 as : put in (19.25.7) and use (19.25.1).
Figure 19.3.5: as a function of and for , . Cauchy principal values are shown when . The function is unbounded as , and also (with the same sign as ) as . As it has the limit . If , then it reduces to . If , then it has the value when , and 0 when . See §19.6(i).
Figure 19.3.6: as a function of and for , . Cauchy principal values are shown when . The function tends to as , except in the last case below. If (), then the function reduces to with Cauchy principal value , which tends to as . See (19.6.5) and (19.6.6). If (), then by (19.7.4) it reduces to , , with Cauchy principal value , , by (19.6.5). Its value tends to as by (19.6.6), and to the negative of the second lemniscate constant (see (19.20.22)) as .

## §19.3(ii) Complex Variables

In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

Figure 19.3.7: as a function of complex for , . There is a branch cut where .
Figure 19.3.8: as a function of complex for , . There is a branch cut where .
Figure 19.3.9: as a function of complex for , . The real part is symmetric under reflection in the real axis. On the branch cut () it is infinite at , and has the value when .
Figure 19.3.10: as a function of complex for , . The imaginary part is 0 for , and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut () it has the value if , and if .
Figure 19.3.11: as a function of complex for , . The real part is symmetric under reflection in the real axis. On the branch cut () it has the value , with limit 1 as .
Figure 19.3.12: as a function of complex for , . The imaginary part is 0 for and is antisymmetric under reflection in the real axis. On the upper edge of the branch cut () it has the (negative) value , with limit 0 as .
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