Analysis

A7

(∑_(k = 0)^∞) Binomial[α, k]*(x^k)

Cauchyprodukt

a[k_, α_] := Binomial[α, k]

cauchysummand = (∑_(j = 0)^k) a[j, α]*
          a[k - j, β]

cauchysummand - a[k, α + β] // FullSimplify

A8

Series[Erf[x], {x, 0, 5}]

Gliedweise Integration

(∑_(k = 0)^∞) 
            (2/(Sqrt[π])) (∫_0^x)((((-(t^2)))^k)/(k!)) \[DifferentialD]t

(2/(Sqrt[π])) (∫_0^x)((((-(t^2)))^k)/(k!)) \[DifferentialD]t

summand = (2/(Sqrt[π])) 
        Integrate[((((-(t^2)))^k)/(k!)), {t, 0, x}, 
          GenerateConditions -> False]

A9

f = ((x y)/((x^2) + (y^2)))

Plot3D[f, {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 50]

grenzwert = Limit[f /. y -> m x, y -> 0]

grenzwert /. m -> 2

grenzwert /. m -> 3

A10

f = (E^(-((x^2) + 2  (y^2))))

Plot3D[f, {x, -2, 2}, {y, -2, 2}, PlotPoints -> 50, 
      PlotRange -> All]

Plot[Evaluate[Table[f /. y -> m x, {m, 1, 5}]], {x, -2, 2}, 
      PlotRange -> All]

ContourPlot[f, {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> 50]

Lineare Algebra

L8

A = {{a, I, 1}, {1, 1, I}, {0, 
          1, -1}}

Inverse[A]

Inverse[A /. a -> 1]

L9

A = {{3, 1, 1}, {2, 2, -1}, {1, -1, 2}, {0, 
          4, -5}}

b = {{3}, {2}, {1}, {0}}

Transpose[{Array[x, {4}]}]

Solve[A . Transpose[{Array[x, {3}]}] == b]

RowReduce[A]

L10

f = (((x^3) + 4  x + 2)/((x^4) - 1))

Apart[f]

Clear[A]

ansatz = 
      (A/(x + 1)) + (B/(x - 1)) + ((C x + D)/((x^2) + 1))

Together[ansatz]

Numerator[Together[ansatz]] - Numerator[f]

gleichungssystem = 
      CoefficientList[Numerator[Together[ansatz]] - Numerator[f], 
        x]

Solve[gleichungssystem == 0, {A, B, C, D}]