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Mathematics Undergraduate

大学数学、複素関数論、テータ関数に関する質問です。 写真のテータ関数の無限積表示(5.24)の式の1行目の形にどうやってしているのかと、命題5.22の(5.26)の証明を教えていただきたいです。

(b) テータ関数 ヤコビは楕円関数論の研究において, 次の級数を導入した。 9(2) = 22(-1)"-!g"-1/2)" sin(2n-1)Tu n=1 2(g/4 sin Tu-g/ sin 3Tu+q^/4 sin 5Tu-…). (5.23) 三 これはヤコビの楕円テータ関数(以下単にテータ関数(theta function))と呼 ばれるものの1つである. limd,(u)/2q'/4=Dsin Tu なので, 0,(u) は sin Tu 9→0 の一種の拡張と見ることができる。 伝統的な記号にならって, 以下 2ミe2miu a=2 q= eir, と書こう.gl<1だから Imr>0である. このとき(5.23)の右辺は TiT 2Tiu 9=e 9 2と(-1)"-1gm-1/2)?_2"-1/2 _2-n+1/2 =iこ(-1)"gm-1/2)°n-1/2 n=1 2i n=-00 = ig4z-1/2 (-1)"g"(n-1)z" n=-00 と書き直すことができる.右辺に3重積公式(5.22)を用いれば, テータ関数 の無限積表示が得られる: 0,(u) = iq'4z-1/2(1-2) II (1-g"2)(1-g"z-')(1-g") n=1. = 2q/4 sin Tu I (1-2g" cos 2Tu+g")(1-g"). 三 (5.24) n=1 命題5.22 0,(u) はuの整関数で 0,(-u) = ー6,(u). (5.25) 0 0(u) = 0 < (m,nEZ). 0,(u+1) = -0, (u), 9,(u+t) = -e-mi(r+2u)9, (u). (5.27) u= m+nT (5.26) 0 + 2u) [証明](5.25),(5.26) は(5.24)から簡単にわかる. また前節の無限積

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English Senior High

高一英語の関係詞の問題です。 一問でも解いてくださると嬉しいです!

標準問題 (2)()に入れるのに最も適切なものを選びなさい。 11 There was something about the man( ) reminded me of. someone I had met before. O that のwhat ③whose のwho (関西学院大) 12 A well-balanced diet is essential for good health. It contains all the substances ) our bodies would not function properly. O because of which のwithout which 3with which O by which 6 on which (中央大) 13 I have been to the place(. ) in her talk. Owhere she spoke of 2which she spoke of ③ that she spoke の of that she spoke (日本大) 14 The girl ( ) used to work for our company. O you were talking to 2 you were speaking ③who were you talking to のwhom you were talking (同志社大) 15 They mentioned two tourist spots, ( ) we have visited as yet. O neither of which ② which of none のwhere of neither の none of which (明治学院大) 16 This isa game( )I am quite ignorant of. Owhich rules 2what rules の the rules of which ④ of whose rules (濁協大) 17 The time will come( ) she will need my help. Owhat のwhich ③when ④whenever (九州共立大) 18 The library was ( ) he met his friend John. Owhich のwho ③ whom のwhen ⑤where (九州産業大) 19 Pat is very talkative. That's( ) I don't like him. O because @why 3 the reason which のwhere (徳島文理大) 20 This is the way( ) we can become acquainted with one another. O how @ in which ③ to which のwhich (明治学院大) 21 Mother does not like( )I clean the house. O how do の how to ③ the way の in which (南山大)

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English Senior High

お願いします

3. The content on the Aspero Designs Web site is protected by copyright law and may not be reproduced - the company's written consent. 1. behind 2. without 3. except 4. before 4. Dr. Hemana and Dr. Wareham, the joint recipients of the Cobalt Research Prize, have known since they were university students in Auckland. 1. other one 2. another one 3. any other 4. each other 5. This afternoon's flights to Barcelona, London, and Rome have all been delayed inclement weather in the destination cities. 1.. as for 2. due to 3. now that 4. only if 6. Patrons who arrive at the theater the show has begun will not be seated until the intermission. 3. along 1. wherever 2. into 4. after 7. The new-product development meeting will be held either in Room 402 in Room 501. 1.. or 2. and 3. not 4. but 8. Printer cartridges can be found in the supply cabinet the file folders. 1.. at 2. from 3. with 4. along 9. Mr. Wong will travel to the management seminar in Singapore on 1.. he 2. himself 3. him 4. his own 10. Market researchers reported that customers were most impressed the Vestra Coffeemaker's delayed-start function. 1.. by 2. beyond 3. for 4. since After December 13 customers will be asked to place their orders online use the mail-order form. 1. So as 2. in case 3. rather than 4. provided that 12. its main competitor, the ergonomic chair offered by Well Designs is lightweight and comes in a variety of colors. 1. In contrast to 2. By way of 3. Instead of 4. So as

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