DLMF
Index
Notations
Search
Help?
Citing
Customize
Annotate
UnAnnotate
About the Project
20
Theta Functions
Properties
20.3
Graphics
20.3
Graphics
20.4
Values at
z
= 0
Figure 20.3.2
(See
in context
.)
Figure 20.3.2:
θ
1
(
π
x
,
q
)
,
0
≤
x
≤
2
,
q
= 0.05, 0.5, 0.7, 0.9. For
q
≤
q
Dedekind
,
θ
1
(
π
x
,
q
)
is convex in
x
for
0
<
x
<
1
. Here
q
Dedekind
=
e
−
π
y
0
=
0.19
approximately, where
y
=
y
0
corresponds to the maximum value of Dedekind’s eta function
η
(
i
y
)
as depicted in Figure
23.16.1
.
ⓘ
Annotations:
Symbols:
η
(
τ
)
: Dedekind’s eta function (or Dedekind modular function)
,
θ
j
(
z
,
q
)
: theta function
,
π
: the ratio of the circumference of a circle to its diameter
,
e
: base of natural logarithm
,
i
: imaginary unit
and
q
: nome
Keywords:
Dedekind’s eta function
,
modular functions
,
relations to other functions
,
relations to theta functions
,
theta functions
Permalink:
http://dlmf.nist.gov/20.3.F2.mag
Encodings:
Magnified png
,
pdf
See also:
Annotations for
§20.3
and
Ch.20