2016年11月9日水曜日

学習環境

解析入門〈1〉 (松坂 和夫(著)、岩波書店)の第1章(数)、1.5(複素数)、問題1.3、1、2.を取り組んでみる。

問1

13i+3 i 2 i 3 =22i ( 23+2i )( 3+4i ) ( 34i )( 3+4i ) = 61 25 + 98 25 i 1( n=4k ) i( n=4k+1 ) 1( n=4k+2 ) i( n=4k+3 )

問2.

| α β |= α β ( α β ) ¯ = α β · α ¯ β ¯ = α α ¯ β β ¯ = α α ¯ β β ¯ = | α | | β |

問3.

2· 10 · 20 2 =40 5 13 2 25 = 130 10

問4.

| α+β | 2 + | αβ | 2 =( α+β ) ( α+β ) ¯ +( αβ ) ( αβ ) ¯ =( α+β )( α ¯ + β ¯ )+( αβ )( α ¯ β ¯ ) = | α | 2 + | β | 2 +α β ¯ + α ¯ β+ | α | 2 + | β | 2 α β ¯ α ¯ β =2( | α | 2 + | β | 2 )

問5.

| ( αβ )+β || αβ |+| β | | | ( αβ )+β | || β || αβ | | α || β || αβ | | α || β || αβ | | | α || β | || αβ |

問6.

| ( α 1 + α 2 +···+ α n )+ α n+1 | | α 1 + α 2 +···+ α n |+| α n+1 | | α 1 |+| α 2 |+···+| α n+1 |

問7.

| αβ 1 α ¯ β | = | αβ | | 1 α ¯ β | = ( αβ ) ( αβ ) ¯ ( 1 α ¯ β ) ( 1 α ¯ β ) ¯ = ( αβ )( α ¯ β ¯ ) ( 1 α ¯ β )( 1α β ¯ ) = | α | 2 α β ¯ α ¯ β+ | β | 2 1α β ¯ α ¯ β+ | α | 2 | β | 2 | α |=1 1α β ¯ α ¯ β+ | β | 2 1α β ¯ α ¯ β+ | β | 2 =1 | β |=1 | α | 2 α β ¯ α ¯ β+1 1α β ¯ α ¯ β+ | α | 2 =1

問8.

1+ | α | 2 | β | 2 ( | α | 2 + | β | 2 ) =( 1 | α | 2 )( 1 | β | 2 )>0 | α | 2 + | β | 2 > | α | 2 + | β | 2 | αβ 1 α ¯ β |<1

問9.

z=c+di z 2 = c 2 d 2 +2cdi c 2 d 2 =a 2cd=b ( c 2 + d 2 ) 2 = a 2 + b 2 c 2 + d 2 = a 2 + b 2 2 c 2 = a 2 + b 2 +a c 2 = a+ a 2 + b 2 2 c=± a+ a 2 + b 2 2 d 2 = a+ a 2 + b 2 2 d=± a+ a 2 + b 2 2 z=±( a+ a 2 + b 2 2 + a+ a 2 + b 2 2 )( b>0 ) z=±( a+ a 2 + b 2 2 a+ a 2 + b 2 2 )( b<0 )

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