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Tanhc function

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Tanhc function

In mathematics, the tanhc function is defined as

Contents

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

    • In terms of other special functions

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

    • Series expansion

    tanhc ⁡ 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 tanh ⁡ ( 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 + O ( z 13 ) )

    Pade approximation

    T a n h c ( 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

    Tanhc function Wikipedia
    (Text) CC BY-SA


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