§19.32 Conformal Map onto a Rectangle
The function
19.32.1
ⓘ
- Symbols:
-
: symmetric elliptic integral of first kind and
: function
- Permalink:
- http://dlmf.nist.gov/19.32.E1
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- Annotations for §19.32 and Ch.19
with 
real constants, has differential
19.32.2
; 
, 
.
; , .
ⓘ
- Symbols:
-
: the ratio of the circumference of a circle to its diameter,
: differential of 
,
: imaginary part,
: phase and
: function
- Referenced by:
- §19.32
- Permalink:
- http://dlmf.nist.gov/19.32.E2
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If
19.32.3
then is a Schwartz–Christoffel mapping of the open upper-half 
-plane
onto the interior of the rectangle in the 
-plane with vertices
19.32.4
ⓘ
- Symbols:
-
: symmetric elliptic integral of first kind,
: imaginary unit and
: function
- Permalink:
- http://dlmf.nist.gov/19.32.E4
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- Annotations for §19.32 and Ch.19
As proceeds along the entire real axis with the upper half-plane on the
right, describes the rectangle in the clockwise direction; hence 
is negative imaginary.
For further connections between elliptic integrals and conformal maps, see Bowman (1953, pp. 44–85).
