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

大学数学の微分学に関する問題です。 先日対数微分法を学習しました。ロピタルの定理は学習していないので、この問題には使用できません。 この問題の解答に対する方針が見えてこないので、お力添えお願いします。

氏名 f(x) 関数f(z) = (? - 1)(arccos z)" + e (-1<r<1)に対して,極限 lim エー0 tan 2』 の値を求めよ。

ア順番に12131であってますか？ わかる方教えてください！！！！ お願いします！

For most of the 20th century, magnetic north was located around the northern parts of molten iron in the earth's core which affect how the earth's magnetic field behaves. What is surprising to scientists is the speed at which the pole has been moving in recent years. magnetic poles, on the other hand, are ( ア ) moving. This fact has several implications poles are located at the northernmost and southernmost ends of the earth. The location of the Shifts in the position of the North Magnetic Pole are nothing new. Since 1831, scientists Earth has two sets of poles: the geographic poles and the magnetic poles. The geographic have been tracking its location. The movements are caused by changes in the flow of swirling for navigation and transportation. Canada. It drified around, moving slowiy north at an ( イ ) speed of approximately 10 MIometers per year. In recent decades. that rate has increased significantly to about 3 KIIometers per year. The North Magnetic Pole is nowmoving away from Canada toward eastern Russia. While scientists can't fully explain how changes in the earth's molten core are affecting the pole's movement, they can map the earth's magnetic field. Doing so allows them to (ウ ) the rate of change over time. This, in turn, provides information on how the magnetic field may shift in the future. Information on changes to the earth's magnetic field is used to produce the World Magnetic Model (WMM). This model is used as the basis for all forms of modern navigation, from how ships move at sea to mapping software in smartphones. ( エ ) inthe model, caused by shits in the real location of the North Magnetic Pole, can seriously impact almost all forms of moderntransportation. The model is updated every five years, but in 2018, during routine checks of the 2015- 2020 model cycle, scientists noticed a problem. Because of the rapidly shifting pole, the WMM was close to being outside the acceptable limits for navigational errors. This forced the release of an updated version of the model. was revised ( オ ) in2020, and the current version is expected to last until 2024. As well as keeping track of the WMM's accuracy, scientists continue investigating why the magnetic ficld is changing so dramatically. This essential work will ensure a safe journey for everyone navigating their way through the world.

アから順番に12131であってますか？ わかる方教えてください！！ お願いします！

例3.次の英文は 「磁極の移動」について述べたものです。(ア) ~ (オ) に入れ る最も適当なものを選択肢から選びなさい。 Earth has two sets of poles: the geographic poles and the magnetic poles. The geographic poles are located at the northernmost and southernmost ends of the earth. The location of the magnetic poles, on the other hand, are ( ア ) moving. This fact has several implications for navigation and transportation. Shifts in the position of the North Magnetic Pole are nothing new. Since 1831, scientists have been tracking its location. The movements are caused by changes in the flow of swirling molten iron in the earth's core which affect how the earth's magnetic field behaves. What is surprising to scientists is the speed at which the pole has been moving in recent years. For most of the 20th century, magnetic north was located around the northern parts of Canada. It drifted around, moving slowly north at an ( イ) speed of approximately 10 kilometers per year. In recent decades, that rate has increased significantly to about 55 kilometers per year, The North Maggnetic Pole is now moving away from Canada toward eastern Russia. While scientists can't fully explain how changes in the earth's molten core are affecting the pole's movement, they can map the earth's magnetic field. Doing so allows them to ( ウ ) the rate of change over time. This, in turn, provides information on how the magnetic field may shift in the future. Information on changes to the earth's magnetic field is used to produce the World Magnetic Model (WMM). This model is used as the basis for all forms of modern navigation, from how ships move at sea to mapping software in smartphones. ( エ ) inthe model、 caused by shifts in the real location of the North Magnetic Pole, can seriously impact almost all foms of modern transportation. The model is updated every five years, but in 2018, during routine checks of the 2015- 2020 model cycle, scientists noticed a problem, Because of the rapidly shifting pole, the WMIM was close to being outside the acceptable limits for navigational errors. This forced the release of an updated version of the model, It was revised (オ ) in2020, and the current version is expected to last until 2024, As well as kecping track of the WMM's accuracy, scientists continue investigating why the magnetic ficld is changing so dramatically, This essential work will ensure a safe joumey for everyone navigating their way through the world.

数Ⅲの数列の極限です。 ａｎやｂｎをなぜ写真のように任意で置くのか分かりません。それぞれなぜ逆数や√で置くのかもわからないです。解説お願いしますm(_ _)m

95 数列 {an}, {b»} において, 次の命題の真偽をいえ。 数列{an}, {b»}において, 次の命題の真偽をいえ。 (2) {anbn}, {an}がともに収束するならば,{b}も収束する。 (1) lim(an-bn)=D 0, liman = α ならば limbn = α (3) lim(an+1- n) = 0 ならば {an}は収束する。 数列の極限の性質(1) 1分 95 1→ 0 1→ 00 →0 式を分ける 数列 {am), {b»}が収束するならば lim(an+ bn) = liman+ lim6,ns limanbn = limanlimbm カ→ 0 1→ 0 れ→ 0 1→ 0 (1) ③ lim(an-bn) = 0 より liman-limbn= 0 合 limb,が収束するとは ガ→ 0 n→ o → 0 誤り 2→ 0 限らないから,誤り。 anbn lim れ→ 0 ln B -a, Bがどのような数でも成り立つか? lim bn → 0 (3) 反例として,lim(an+1- an) =0 であるが liman = o となる {an}を考える。 第→ 00 不定形 o - o で0に収束< Action》数列の収束の判定は, 収束する数列の和 差 積·商を考えよ (1) limbn = lim{an- (an-bn)} = liman lim(an- b) {b}の収束,発散がわか らないから,単純に lim(an-bn) 1→ 0 n→ 0 n→ 0 c0- =α-0 = a したがって,この命題は真である。 = lima,- limb, ガ→ 00 とはできない。 an bn = nとすると n |lima, = 0 のとき #→ 0 limanba 11 Tim n→o n liman lim n→ 0 n anbn limb, = lim B = 0 n→ 0 n→ 0 0 1→ o とはできないから, lima, = 0 となる例を考 よって, 数列 {an6,}, {an}はともに収束する。 ところが, limbn limn =8 となり,数列 {bn} は発散 える。 2→0 8t4 する。したがって, この命題は偽である。 反例,すなわち {an+1-an}は0に収束 るが{an}が発散する色 をさがす。 an = Vとすると m(an+1-4m) =Dlim(/n+1-/n) O- 1 = 0 lim 2→ 0 n ところが, liman = lim n=8 となり, 数列 {am} は発 n→ 0 2→ o 敗する。したがって, この命題は偽である。 Un R ならば lim bn B →0 2

nを極限に飛ばして、x^2＋2になったのですが、そこからどうやってa,bの値を求めればいいか分かりません。教えてください

【1】 a,bは定数で, aキー1,0とする。 7+2 + 2c" + bx-a+3 関数f(x)= lim -が x>0で微分可能であるとき, 0"+a n→0 a,bの値を求めよ。 1 a= 3 2

この２問のやり方を教えていただきたいです🙇🏻 「関数の極限」なんですけど他の問題は普通に代入したり＝０になる場合のみ変形したりする問題などは分かります！

1| (x-2)?」 バキ (xリ x+2 245(1) lim (2) lim{1- ー(x-1)? x→1 x→2 +3

わかんないので教えてください🙏

11 Do you think you are grown-up? om Child ぐお来の人大1O You have probably made various friends since vo. entered high school. What kind of people are they? Some of them may seem more grown-up than you. nAp) They are often calm, hardworking, and confident. Others may seem more childish. They are often short- tempered, impatient, and self-centered. Which kind of person are you? Are you grown-up red for your age? How do your friends see you? You will mature as you have a variety of experiences. Little by dar] little, you will climb the long ladder to adulthood. Now * Althud] is the time when you are taking your first steps. 10 00 1.

答えだけでいいので教えてください＜(_ _)＞

第17章 主語と動詞 例 題 154 The number of odrug-related crimes aare aincreasing yearly. 文指機 Both Sophia and I ( ) disappointed by your behavior. のbe 155 Dare 2am 3is 156 Neither Emma nor I ( ) able to attend the event. Dis 2am 3are のhave 157 A woman owith two babies in her arms oare sitting on the bench. 国文指摘 演習 ロ 1. The number of students in the French class ( ) limited to thirty. (東京造形大) Ois 2are 3have のhas ロ 2. A number of students ( ) come to the event. (亜細亜大) Ohave 2are 3has were 3. Both of my brothers ( )married. (芝浦工業大) Dis 2are Obe was ロ 4. Either my boss or I ( ) supposed to attend the meeting next month. Ois @am 3are Owill (東京電機大) ロ 5. Neither Tina nor I ( ) to blame. (東京工芸大) Oam のis 3are Ocan ロ 6. Not only you but also your wife ( ) invited to the company's event. Dwere 2are 3be Ois (園季院大)

この答えになる途中式を教えていただきたいです。

*具体例2 dae a? の数値積分(厳密解は1/3) 1-Ar 1-A¢ I:= lim,E「(r)Ar, I(Ar) := > f (x)A» とf(x)A Ar→0 =0 C=0 I(Ar = 0.100) = 0.285 I(Ac = 0.010) I(Ar = 0.001) = 0.333 ~1/3 ← = 0.328 近似的limit 値を得た