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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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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1. The Milltown Cinema's outdoor cafe will be closed the winter months. 1. about 2. out of 3. next to 4. during 2. The department-wide changes suggested unanimous approval from company management. Ms.Juntasa received 1. in 2. of 3. by 4. as 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. 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. 1. wherever 2. into 3. along 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 delayed-start function. 1.. by the Vestra Coffeemaker's 2. beyond 3. for 4.. since

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6. Patrons who arrive at the theater 1.. wherever the show has begun will not be seated until the intermission. 2. into 3. along 4. after 7. The new-product development meeting will be held either in Room 402 in Room 501. 1. 2.. and 3. not 4. but or 8. the file folders. 4. along Printer cartridges can be found in the supply cabinet 1.. at 2. from 3. with 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 delayed-start function. 1.. by - the Vestra Coffemaker's 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. ts 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 13. entering the main lobby. 3. Guests are asked to register at the front desk 1. upon 2. about in order to 4. whoever

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The Milltown Cinema's outdoor cafe will be closed the winter months. 1. about 2. out of 3. next to 4. during 2. The department-wide changes suggested Ms. Juntasa received unanimous approval from company management. 1. in 2. of 3. by 4. as 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 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

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The Milltown Cinema's outdoor cafe will be closed the winter months. 1. about out of 3. next to 4. during 2. The department-wide changes suggested unanimous approval from company management. 1.. in Ms. Juntasa received 2. of 3. by 4. as 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 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. 1.. wherever 2. into 3. along 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. 3. 1. at 2. from 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 wayof 3. Instead of 4. So as

対数微分法を使ってこの関数の導関数を求めるとはどういうことでしょうか？解答までの過程も含めて教えてください。

8. Use logarithmic differentiation (封教微分法) to find the derivative of the function y= xVx? +1 x>0. 2/3 ラ (x+1)?3

数列の極限です。数Bの数列と同様に数列の極限でもｎは自然数というのが暗黙の了解なのですか？

CHA から 1+に 答 1 COS nπ -1ScoS nnS1であるから 1 n 4各辺をnで割る。 n n VI VI

この教科書のレベルはどのくらいですか教えください この教科書でどのくらいのレベルの大学まで対応できますか？

1 On 10 February 2009, at a height of about 800 kilometers above Siberia, an American satellite collided the first such height [háit] satellite [séetalait] collide(d) [kaláid(id)] with an old Russian satellite. It was collision [kaligan] collision in the history of space development. As a result, fragment(s) [fráegmant(s)) debris [dabri:] more than 1,000 fragments of debris were scattered into space. 2 The image above shows the vast amount of space debris in orbit around Earth. Approximately 22,000 vast [váest] orbit [5:rbat] approximately [aprá:ksamatli) objects larger than 10 centimeters across are floating around Earth. Of these, about 16,000 are from known 10 considering [kansidarig) artificial [a:rtafijal] currently [ks:rantli] operation [a:paréifon] Considering that there are only about 1,000 artificial satellites currently in operation, the amount of Sources. space debris is astonishing. This space debris is not only due to the collision of satellites. For example, when rockets reach space, they s 15 leave behind surplus engines and fuel tanks. These objects remain in orbit as space debris. In addition, surplus s5:rplas] there are tools that astronauts have dropped while tool(s) [t:l(z)) astronaut(s) [astrand:t(s) aluminum [ala:manom per|par] working outside. Even a one-centimeter aluminum ball. when orbiting at a speed of around 10 kilometers per 0 bullet [bálat] second, is far more powerful than a bullet from a gun. gun [gán]

多様体の接空間に関する基底定理の証明です。g(q)=∫〜と定義した関数を微積分学の基本定理を用いながら変形してg(q)=g(0)+∑gᵢuⁱと導出するのですが、これがうまくいきません。 自分は、g(q)の式をまず両辺tで微分して、次に両辺uⁱで積分して、最後に両辺tで積分... 続きを読む

12. Theorem.If{ = (x', , x") is a coordinate system in M at p, then its coordinate vectors d, lp, …… 0,l, forma basis for the tangent space T,(M); and D= E(x) 。 i=1 for all ve T(M). Proof. By the preceding remarks we can work solely on the coordinate neighborhood of G. Since u(c) = Othere is no loss of generality in assuming ど(p) = 0eR". Shrinking W if necessary gives E(W) = {qe R":|q| < } for some 8. Ifg is a smooth function on E(W) then for each 1 <isndefine og (tq) dt du g(9) = for all qe {(W). It follows using the fundamental theorem of calculus that g= g(0) + E&,u' on (W). Thus if fe &(M), setting g = f。' yields f= f(P) + Ex on U. Applying d/ax' gives f(p) = (f /0x)(P). Thus applying the tangent vector e to the formula gives (f) = 0+ E(x'(p) + E Ap)u(x) = E(Px). ず ax Since this holds for all f e &(M), the tangent vectors v and Z Ux') d,l, are equal. It remains to show that the coordinate vectors are linearly independent. But if ) a, o.l, = 0, then application to x' yields dxi 0=24 (P) = 2q d」= 4. In particular the (vector space) dimension of T,(M) is the same as the dimension of M.