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Mathematics Undergraduate

複素数の問題です。 全て解いてほしいです。 特に問題4の解説をよろしくお願いします。

問 ■複素平面と極形式 題 複素数zは:=Rez+ i Imz と書くことができ、実部 Re z をx座標、虚部 Im:をy座標に見立てることで、 ガ ウ こを2次元平面上の1点として捉えることができる。この平面を複素(数)平面ないしGauss 平面と呼ぶ。 一方、ある複素数zを、二つの実数r,e(ただしr>0に制限す る)を用いて Im ミ=ree という形で表わしたものを:の極形式表示と呼ぶ。e の逆数は -1 Im:=rin 1 で定義する。 er Imz 問[]()r= |, tan @ = が成り立つことをそれぞれ示せ。 Rez (i) 逆数の定義に基づいて (e")= e-t0 であることを示せ。 Re Rez=r このようにこの絶対値であるrは複素平面における原点(0+ 0i) から、までの距離を表わし、0は原点とこを結ぶ線分が実軸となす 角を表わす。はarg z とも書き、偏角 (argument)(物理や工学で はしばしば位相(phase))と呼ぶ。原点の周りを一周しても同じ点 に戻ってくることから、0には 2x ラジアン= 360度の整数倍の不 定性がある。また、0+0iの偏角は定義されない。 図1 複素平面。 偏角と加法定理 絶対値が1の二つの複素数 Im 21= COs # +isin @, 2= cos #,+i sin @。 を考える。ここで0,,02 は実数とする。 問 [2]() 積22 を計算し、三角関数の加法定理とオイラーの公 式を用いて極形式表示に直せ。また、同様にして商z/zz = zi の極形式表示も求めよ。(i) 21,22の複素平面における表示を図2 とする。このとき、積」みと商z/を複素平面に図示せよ。 0.5 Re -10 -0.5 0.5 21= e,22= e であったから、小間 (i) のとくに積の方の結 果から、次の基本的な指数法則が成り立つことが理解できる: 基本的な指数法則 -0.5 実数,に対してelh el = e(h+h)が成り立つ。 図2 と2の複素平面における表示。 また、小間(i) の結果から、22= e' hを掛けることで」から偏 角がだけ反時計回り方向に回り(角度が+)、2で割ることで 2」から偏角はだけ時計回り方向に回る(-)ことが納得できる。

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TOEIC・English Undergraduate

Could anyone check if my answers are correct

ad been' performing for two hours when the power went off. buy / wasn't buying the dress. 2 correct the sentences. bake a cake for my friend's birthday yesterday. Ibaked a cake for.my.friend's bithday. yesterday. 2. Deyet feet energetic right now? 3. My teacher didn't gave me back my test. Li you te l 4. Roberto had went to bed beforel came home. 5. My brother had-stept when I called. 6. Stefan is very ambitious. He's sttdies to be a doctor. en kd skepiny オ 3 Complete the message. Use the Correct form of the verbs in parentheses. en Search Join Home Create Messenger Mina ■ 0 arch Mina Kim Hey Kim! Options I'm so happy that you got in contact! we left high school. T haven't heard from you since a Search conversation Edit profile about me … ? Well, after high school | ' went What can I tell you Notifications (go) to college for four years and studied marine biology. While | (learn) about the ocean and marine life, I decided that my ambition was to work in an aquarium. At first, I wasn't confident that| (can) get that kind of job because it's so popular. But I was 3 patient and worked hard, and nowl'm working at an aquarium in Toronto. I's a litle far from my house, sol4_r0e day-it takes about 50 minutes. I'm really happy there. I especially love working with the seals-they're so affectionate! (drive) there by car every lenakl cdecide) 98gn Ma How about you?I heard from Maria that to move to France before you finished college. __// there now? Are 5 you (live) (have) kids? Please let me 7 married? you know.I really want to hear about what ©_ 90lhg on_ (go on) in your life right now. Hope to hear from you soon! Love, Mina PS Here's a picture of me with a coworker!

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English Senior High

急ぎです🙏🏻🙏🏻高2の英語の問題です。1枚目を参考に2枚目の質問に答える問題なのですが、書き方や文法があっているか教えて頂きたいです。

Strongest! | Lesson 1 Part 1 challenge [elinds] の “I know a successful player, 1) who's been trying to achieve. a Roger Federer [rá()dgar fedarar] wheelchair [hwi:ltfear] rank [repk] greater challenge than mine.” Roger Federer, one of the world's 5 best tennis players, showed great major (形) [mérd3ar] 第 official [afjal] ア respect to a Japanese tennis player. His name is Shingo Kunieda. He Paralympics [paèralimpiks] Roger Federer is one of the top wheelchair tennis players, and has tournament [ttarnamant, t3:r-] except [ksépt] bounce [báuns] G-1) been ranked first in the world for many years. 2) Wheelchair tennis is one of the major wheelchair 10 sports in the world. It has the status of an official accurately [ékjaratli] の technique [tekni:k]の sport in the Paralympics, and has as many tournament G-1 games as regular tennis does. Its rules are almost pay [pé] the same as those of regular tennis, except for one 15 thing. The players can let the ball bounce up to two !)TRIVIA times before returning it. Wheelchair tennis requires technique「技術」 専門技術·技法,および スポーツ·芸術などの技術· テクニックを表す。 一方、 skill は熟練した技術·能 カ·手腕などを表す。 players to have the skills to control the wheelchair quickly and accurately, as well as the techniques of tennis. Players also need to pay careful attention 20 to their body condition while they are playing. *. uie sanme as A His cellphone is the same as mine. 14. except for A Except for one mistake, you report is excellent. 15. up to A You can borrow up to three books in this library. 19. pay attention

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