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Tips for Maxima command
Category: ICT
Published: 2021
#2121a

compiled by Kanzo Kobayashi
up 21z31
Index
Key
 ; ; if文; ; 関数の定義 ; 行列式; 極限値; 式の展開; 数列の和; 数列の積; 三角関数; 多項式; テイラー展開; ; 部分分数; ; ; 連立方程式; ; ; ; ;
Key
Maxima
Remarks

>Top <A>:

<A>:

<A>:

>Top <B>:

<B>:

<B>:

>Top <C>:

<C>:

<C>:

>Top <D>:

  1. Differentiate 微分
  2. Definition of function 関数定義

<D>:

  1. diff(a*x^3+b*x^2+c*x+d,x,1);→$3a^2+2bx+c$
    1. diff((x^2+a^2)/(x^2-a^2),x,1);→$\frac{2x}{x^2-a^2}-\frac{2x(x^2+a^2)}{(x^2-a^2)^2}$
    2. diff(sqrt(x^2+1),x);→$\frac{x}{\sqrt{x^2+1}}$
    3. diff((exp(3*x)+exp(-2*x))/2,x);→$\frac{3e^{3x}-2e^{-2x}}{2}$
    4. diff(2^x,x);→$2^x\log 2$
    5. diff((x+1)/(x+2)^(4/3),x);→
      $\frac{1}{(x+2)^{\frac{4}{3}}}-\frac{4(x+1)}{3(x+2)^{\frac{7}{3}}}$
    6. diff(log(x+sqrt(x^2+1)),x);
      ratexpand(%); →$\frac{1}{\sqrt{x^2+1}}$
    7. diff(sin(x)^2,x);→$2\cos x\sin x$
      trigreduce(%); →$\sin 2x$
    8. diff(atan(x),x);→\frac{1}{x^2+1}
  2. f(x):=x^2+2*x+1; →$f(x):=x^2+2x+1$
    factor(f(x)); →$(x+1)^2$

<D>:

>Top <E>:

  1. expand 式の展開

<E>:

  1. expand((x+y)^4); →$y^4+4xy^3+6x^2y^2+4x^3y+x^4$
    factor(%); →$(y+x)^4$

<E>:

>Top <F>:

<F>:

<F>:

>Top <G>:

<G>:

<G>:

>Top <H>:

  1. Hyperbolic function

<H>:

Hyperbolic Function Trigonometric Funtion
$\sinh x=\frac{e^x-e^{-x}}{2}$ $\sin x=\frac{e^{ix}-e^{-ix}}{2i}$
$\cosh x=\frac{e^x+e^{-x}}{2}$ $\cos x=\frac{e^{ix}+e^{-ix}}{2}$
$\tanh x=\frac{\sinh x}{\cosh x}$ $\tan x=\frac{\sin x}{\cos x}$
$\sinh(-x)=-\sinh x$ $\sin(-x)=-\sin x$
$\cosh(-x)=\cosh x$ $\cos(-x)=\cos x$
$\tanh(-x)=-\tanh x$ $\tan(-x)=-\tan x$
$\cosh^2x-\sinh^2x=1$ $\cos^2x+\sin^2x=1$
$1-\tanh^2x=\frac{1}{\cosh^2x}$ $1+\tan^2x=\frac{1}{\cos^2x}$
$e^x=\cosh x+\sinh x$ $e^{ix}=\cos x+i\sin x$
$\sinh(\alpha±\beta)=\sinh\alpha\cosh\beta
±\cosh\alpha\sinh\beta$		 $\sin(\alpha±\beta)=\sin\alpha\cos\beta
±\cos\alpha\sin\beta$
$\cosh(\alpha±\beta)=\cosh\alpha\cosh\beta
±\sinh\alpha\sinh\beta$		 $\cosh(\alpha±\beta)=\cosh\alpha\cosh\beta
\mp\sinh\alpha\sinh\beta$
$\tanh(\alpha±\beta)=\frac{\tanh\alpha±\tanh\beta}
{1\mp\tanh\alpha\tanh\beta}$		 $\tan(\alpha±\beta)=\frac{\tan\alpha±\tan\beta}
{1\mp\tan\alpha\tan\beta}$
$\lim_{x\to 0}\frac{\sinh x}{x}=1$ $\lim_{x\to 0}\frac{\sin x}{x}=1$
$\lim_{x\to 0}\frac{\cosh x-1}{x}=0$ $\lim_{x\to 0}\frac{\cos x-1}{x}=0$
$\lim_{x\to 0}\frac{\tanh x}{x}=1$ $\lim_{x\to 0}\frac{\tan x}{x}=1$

<H>:

>Top <I >:

  1. if, then, else文

<I>:

  1. s(x):=if x<0 then 0 else 1;
    1. h(x):=if x<0 then 0 elseif x=0 then 1/2 else 1;

<I>:

  1. If 条件式1 then 処理1 elseif 条件式2 then 処理2 ... elseif 条件式n then 処理n else 処理0

>Top <J>:

<J>:

<J>:

>Top <K>:

<K>:

<K>:

>Top <L>:

  1. Limit

<L>:

  1. limit((1+x)^(1/x),x,0);→e

<L>:

  1. $\displaystyle\lim_{\to +0}(1+x)^{\frac{1}{x}}
    → e$

>Top <M>:

  1. Matrix 行列式

<M>:

  1. A: matrix([1,2],[3,4]);
    1. B: matrix([a,b],[c,d]);
      C: A+B;
      D: A.B;
      E: 3*B;
      F: A^^2;
    2. kill (all);
      C: matrix([a,b,c],[d,e,f]);
      transpose(C);
    3. A: matrix([1,2,3],[4,5,6],[7,8,9]);
      echelon(A);
      rank(A);
    4. B: matrix([1,2,3],[4,5,6],[7,8,0]);
      invert(B);
    5. C: matrix([a,b],[c,d]);
      invert(C);
    6. E: matrix([a,b,c],[d,e,f],[g,h,i]);
      row[E,1];

<M>:

  1. 行列式
    1. 加算
      乗算
      スカラー倍
      冪乗
    2. 転置行列
    3. 階段行列 (echelon)
    4. 逆関数
    5. 逆関数
    6. 行の取り出し

>Top <N>:

<N>:

<N>:

>Top <O>:

<O>:

<O>:

>Top <P>:

  1. Partial fraction 部分分数展開 ･
    式の簡略化
  2. Polynominals 多項式
  3. product

<P>:

  1. partfrac(1/(x^2*(x^2+1)), x);→$\frac{1}{x^2}-\frac{1}{x^2+1}$
    1. partfrac(x/(x+1),x);→$1-\frac{1}{x+1}$
      ratsimp(%);→\frac{x}{x+1}
  2. poly:x^2-x-12;
    1. solutions:solve(poly=0,x);
    2. map(lambda([eq],lhs(eq)-rhs(eq)),solutions);
  3. product(sin(k*%pi/4),k,1,3); →$\frac{1}{2}$

<P>:

  1. $\frac{1}{x^2(x^2+1)}$
  2. x^2-x-12
    1. [x=-3, x=4]
    2. [x+3, x-4]
  3. $\Pi$

>Top <Q>:

  1. Quotation

<Q>:

  1. solve([x+y=1, x^2+y^2=2], [x,y]);→$[[x=\frac{1-\sqrt{3}}{2}, y=\frac{1+\sqrt{3}}{2}], [x=\frac{1+\sqrt{3}}{2}, y=\frac{1-\sqrt{3}}{2}]]$

<Q>:

  1. $\cases{x+y=1\\x^2+y^2=2}$

>Top <R>:

<R>:

<R>:

>Top <S>:

  1. Simultaneous equations
  2. solve
  3. sum; nusum

<S>:

  1. eq1: [x+2*y+3*z=4, 5*x+6*y+7*z=8, 9*x+10*y+11z=0];
  2. solve(x^3-2*x+1=0, x); →$[x=-\frac{\sqrt{5}+1}{2}, x=\frac{\sqrt{5}-1}{2}, x=1]$
    1. solve(a*x^2+b*x+c=0, x);→$x=\frac{-b±\sqrt{b^2-4ac}}{2a}$
    2. solve([x^2+y^2=10,y=2*x-5],[x,y]);
      →[[x=1,y=-3],[x=3,y=1]]
    3. solve(x^2+1=0); →[x=-%i, x=%i]
  3. sum(2*k+1, k, 1, n),simpsum; →n2+2n
    1. nusum(k,k,1,n); →$\frac{n(n+1)}{2}$
    2. nusum(k^2,k,1,n);→$\frac{n(n+1)(2n+1)}{6}$
    3. nusum(a*r^(k-1),k,1,n); →$\frac{a(r^n-1)}{r-1}$
    4. sum(1/x^2,x,1,inf), simpsum=true; →$\frac{\pi^2}{6}$
  4. a

<S>:

  1. $\cases{x+2y+3z=4\\5x+6y+7z=8\\9x+10y+11z=0}$
  2. $x^3-2x+1=0$
    1. $ax^2+bx+c=0$
  3. $\displaystyle\sum_{k=1}^{n}(2k+1)$
    1. $\displaystyle\sum_{k=1}^{n}k$
    2. $\displaystyle\sum_{k=1}^{n}k^2$
    3. $\displaystyle\sum_{k=1}^{n}ar^{k-1}$
    4. $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}$

>Top <T>:

  1. Tayloer Sereies
  2. Trigonometric function

<T>:

  1. taylor(%e^x,x,0,5); → $\frac{1}{120}x^5+\frac{1}{24}x^4+
    \frac{1}{6}x^3+\frac{1}{2}x^2+x+1$
    1. taylor(sin(x), x, 0, 5);→
      $\frac{1}{120}x^5-\frac{1}{6}x^3+x+\cdots$
    2. taylor(cos(x), x, 0, 5);→
      $\frac{1}{24}x^4−\frac{1}{2}x^2+1$
    3. taylor(log(1+x), x, 0, 3);→
      $\frac{1}{5}x^5−\frac{1}{4}x^4+\frac{1}{3}x^3
      -\frac{1}{2}x^2+x+\cdots$
  2. trigexpand(sin(%alpha+%beta)); →$\cos\alpha\sin\beta+\sin\alpha\cos\beta$
    1. sin(2*%alpha);→$2\cos\alpha\sin\alpha$
    2. cos(2*%alpha);→$\cos^2\alpha-\sin^2\alpha$
    3. tan(2*%alpha);→$\frac{2\tan\alpha}{1-\tan^2\alpha}$
    4. sin(3*%alpha);→$3\cos^2\alpha\sin\alpha-\sin^3\alpha$
    5. cos(3*%alpha);→$\cos^2\alpha-\sin^2\alpha$
    6. tan(3*%alpha);→$\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha}$

<T>:

  1. $e^x=1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+
    \frac{1}{4!}x^4+\frac{1}{5!}x^5+\cdots$
    1. $\sin x=x-\frac{1}{3!}x^3
      +\frac{1}{5!}x^5-\frac{1}{7!}x^7
      +\cdotss $
    2. $\cos x=1-\frac{1}{2!}x^2
      +\frac{1}{4!}x^4-\frac{1}{6!}x^6
      +\cdots $
  2. Trigonometric:
    1. $2\sin\alpha\cos\alpha$
    2. a
    3. a
    4. $3\sin\alpha-4\sin^3\alpha$
    5. $4\cos^3\alpha-3\cos\alpha$
    6. a

>Top <U>:

<U>:

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>Top <V>:

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>Top <W>:

<W>:

<W>:

>Top <X>:

<X>:

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>Top <Y>:

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>Top <Z>:

<Z>:

<Z>:
Key
Maxima
Remarks
Pinyin
Comment
  • For a long time, I used paper, pcncils, rulers and a compass to study mathematics. This time my computer with Maxima software will participate in them.
  • 今まで長い間、紙と鉛筆、定規とコンパスを使って数学を勉強してきた。今回はMaximaソフトを使ってコンピュータが参加することになる。

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