The Use of Integral Transforms |
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Page 379
... ( cosha + cos ẞ ) P- + it ̄m ( cosh α ) e ± tß ± imó ( 7-3-23 ) where to ensure the property ( 7-3-22 ) we must take m to be an integer . Putting m = 0 , we deduce the axisymmetric solutions ( α , ẞ ) = √ ( cosha + cos ẞ ) P_ ++ it ...
... ( cosha + cos ẞ ) P- + it ̄m ( cosh α ) e ± tß ± imó ( 7-3-23 ) where to ensure the property ( 7-3-22 ) we must take m to be an integer . Putting m = 0 , we deduce the axisymmetric solutions ( α , ẞ ) = √ ( cosha + cos ẞ ) P_ ++ it ...
Page 385
... cosha + cos ß π √ ( cosha + cos ẞ ) - ∞ cosh ( BT ) P- ++ it ( cosh α ) dɩ cosh ( πτ ) 1 -π < В < л . ( 7-4-24 ) √2 ( cosha + cos ẞ ) ' Some special cases of the use of equation ( 7-4-13 ) arise frequently so we shall derive them ...
... cosha + cos ß π √ ( cosha + cos ẞ ) - ∞ cosh ( BT ) P- ++ it ( cosh α ) dɩ cosh ( πτ ) 1 -π < В < л . ( 7-4-24 ) √2 ( cosha + cos ẞ ) ' Some special cases of the use of equation ( 7-4-13 ) arise frequently so we shall derive them ...
Page 388
... ( cosha + cosh t ) G. ( t ) dt 。√ ( cosha + cosh t ) ( 7-4-34 ) where G. ( t ) denotes the Fourier cosine transform of the function g ( t ) . 7-5 FOCK'S THEOREMS We shall begin by finding the form of the operator o . Suppose that Þo ̄1 ...
... ( cosha + cosh t ) G. ( t ) dt 。√ ( cosha + cosh t ) ( 7-4-34 ) where G. ( t ) denotes the Fourier cosine transform of the function g ( t ) . 7-5 FOCK'S THEOREMS We shall begin by finding the form of the operator o . Suppose that Þo ̄1 ...
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a₁ analytic b₁ Bessel function boundary conditions boundary value problem coefficients consider constant converges convolution theorem cosech cosh deduce defined by equation definition denotes derive diffusion equation dx² easily shown equivalent example Find the solution finite Fourier cosine transform Fourier sine transform Fourier transform function f(x half-plane Hankel transform hence initial conditions integral equation interval inversion formula inversion theorem kernel Laplace transform Laplace's equation lemma Mellin transform method obtain the relation obtain the solution operator partial differential equation polynomial positive integer positive number Prob Prove R₁ real line respect result right-hand side satisfies the conditions satisfying the boundary Similarly simple sinh Stieltjes transform tanh tends to zero variable write written πτ ди дх