Rahul Sharma (Editor)

Tanc function

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Tanc function

In mathematics, the Tanc function is defined as

Contents

Tanc ⁡ ( z ) = tan ⁡ ( z ) z
Imaginary part in complex plane
  • Im ⁡ ( tan ⁡ ( x + i y ) x + i y )
    • Real part in complex plane
  • Re ⁡ ( tan ⁡ ( x + i y ) x + i y )
    • absolute magnitude
  • | tan ⁡ ( x + i y ) x + i y |
    • First-order derivative
  • 1 − ( tan ⁡ ( z ) ) 2 z − tan ⁡ ( z ) z 2
    • Real part of derivative
  • − Re ⁡ ( − 1 − ( tan ⁡ ( x + i y ) ) 2 x + i y + tan ⁡ ( x + i y ) ( x + i y ) 2 )
    • Imaginary part of derivative
  • − Im ⁡ ( − 1 − ( tan ⁡ ( x + i y ) ) 2 x + i y + tan ⁡ ( x + i y ) ( x + i y ) 2 )
    • absolute value of derivative
  • | − 1 − ( tan ⁡ ( x + i y ) ) 2 x + i y + tan ⁡ ( x + i y ) ( x + i y ) 2 |

    • In terms of other special functions

  • Tanc ⁡ ( z ) = 2 i K u m m e r M ( 1 , 2 , 2 i z ) ( 2 z + π ) K u m m e r M ( 1 , 2 , i ( 2 z + π ) )
  • Tanc ⁡ ( z ) = 2 i HeunB ⁡ ( 2 , 0 , 0 , 0 , 2 i z ) ( 2 z + π ) HeunB ⁡ ( 2 , 0 , 0 , 0 , 2 ( i / 2 ) ( 2 z + π ) )
  • Tanc ⁡ ( z ) = W h i t t a k e r M ( 0 , 1 / 2 , 2 i z ) W h i t t a k e r M ( 0 , 1 / 2 , i ( 2 z + π ) ) z

    • Series expansion

    Tanc ⁡ z ≈ ( 1 + 1 3 z 2 + 2 15 z 4 + 17 315 z 6 + 62 2835 z 8 + 1382 155925 z 10 + 21844 6081075 z 12 + 929569 638512875 z 14 + O ( z 16 ) ) ∫ 0 z tan ⁡ ( x ) x d x = ( z + 1 9 z 3 + 2 75 z 5 + 17 2205 z 7 + 62 25515 z 9 + 1382 1715175 z 11 + 21844 79053975 z 13 + 929569 9577693125 z 15 + O ( z 17 ) )

    Pade approximation

    Tanc ⁡ ( z ) = ( 1 − 7 51 z 2 + 1 255 z 4 − 2 69615 z 6 + 1 34459425 z 8 ) ( 1 − 8 17 z 2 + 7 255 z 4 − 4 9945 z 6 + 1 765765 z 8 ) − 1

    References

    Tanc function Wikipedia
    (Text) CC BY-SA


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