The Construction and Applications of Modular Equations

We introduce also two parametrizations hk,n and h'k,n of the theta-function 4q . For q=e2piz , Im z > 0, define 4q : =fq,q=n =-infinityinfinityqn2 =:q30,2z , where q3 is classical theta function. For any positive real numbers n and k, define hk,n=4e-p n/k k1/44e-p nk=q 30,-n/k k1/4q30, -nk and h'k,n=4 -e-2pn/k k1/44-e-2p nk =q30,1+2-n/ kk1/4q 30,1+2-nk . We show how they are connected with rk,n, r'k,n and Ramanujan-Weber class invariants and give some new theta-function identities. In addition, we establish some formulas for hk,n and h'k,n by using those theta-function identities and find some values of hk,n and h'k,n . Finally, we offer explicit formulas for 4-e-np and 4e-np for any positive real numbers n. We give some examples.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Yi, Jinhee

Contributors dc:contributor

• Berndt, Bruce C.

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3017261

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86780