[PDF] [PDF] Fourier Sine Transforms - EqWorld

[PDF] Fourier Sine Transforms - EqWorld

Fourier Sine Transforms: Expressions with Hyperbolic Functions No Original Sine transform, ˇfs(u) = ∫ ∞ 0 f(x) sin(ux) dx 1 1 sinh(ax) , a > 0 π 2a tanh

A Tables of Fourier Series and Transform Properties

Table A 2 Properties of the continuous-time Fourier transform x(t) = 1 2π ∞ ∫ Table B 1 The Fourier transform and series of basic signals tanh x = sinh x

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Signals Systems - Reference Tables 1 Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform Р ¥ ¥-

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5 jan 2020 · http://www math umn edu/˜garrett/m/real/notes 2019-20/08e Fourier transform deserve to be in one's bestiary of known Fourier transforms:

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The following expressions hold for x >0 tanh x is epanded to Fourier series for x >0 as follows Differentiating both sides of (2 0) with respect to x repeatedly, we 

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Auxiliary Sections > Integral Transforms > Tables of Fourier Sine Transforms > Fourier Sine Transforms:

Expressions with Hyperbolic Functions

Fourier Sine Transforms: Expressions with Hyperbolic Functions No

Original function

,f(x)

Sine transform

fs(u)=Z 1 0 f(x)sin(ux)dx 1 1 sinh(ax),a>0

2atanh¡1

2

¼a-1u¢

2 x sinh(ax),a>0

2sinh¡1

2

¼a-1u¢

4a2cosh2¡1

2

¼a-1u¢

3 1 x e-bxsinh(ax),b>|a| 1 2 arctan³2au u

2+b2-a2´

4 1 xcosh(ax),a>0 arctan

£sinh¡1

2 5

1 -tanh¡1

2 ax¢,a>0 1 u asinh¡¼a-1u¢ 6 coth ¡1 2 ax¢- 1,a>0 a coth¡¼a-1u¢-1 u 7 cosh(ax) sinh(bx),|a|2bsinh¡¼b-1u¢ cos

¡¼ab-1¢+cosh¡¼b-1u¢

8 sinh(ax) cosh(bx),|a|¼ab-1¢sinh¡1 2

¼b-1u¢

cos

¡¼ab-1¢+cosh¡¼b-1u¢

References

Bateman, H. and Erd

´elyi, A.,Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954.

Ditkin, V. A. and Prudnikov, A. P.,Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.

Polyanin, A. D. and Manzhirov, A. V.,

Handbook of Integral Equations

, CRC Press, Boca Raton, 1998. Fourier Sine Transforms: Expressions with Hyperbolic Functions

Copyright

c°2005 Andrei D. Polyaninquotesdbs_dbs17.pdfusesText_23