f (t) = A cos (2πωc t + B sin (2πωm t)) ࡾ֯ؔͷՃ๏ఆཧ f (t) = A cos (2πωc t) cos (B sin (2πωm t)) −A sin (2πωc t) sin (B sin (2πωm t)) cos (a + b) = cos a cos b − sin a sin b
f (t) = A cos (2πωc t) cos (B sin (2πωm t)) −A sin (2πωc t) sin (B sin (2πωm t)) cos (z sin θ) = J0 (z) + 2 ∞ ∑ k=1 J2k (z) cos (2kθ) sin (z sin θ) = 2 ∞ ∑ k=0 J2k+1 (z) sin ((2k + 1) θ) Olver, Frank W. NIST handbook of mathematical functions. Cambridge New York: Cambridge University Press NIST, 2010. p.226
cos (z sin θ) = J0 (z) + 2 ∞ ∑ k=1 J2k (z) cos (2kθ) sin (z sin θ) = 2 ∞ ∑ k=0 J2k+1 (z) sin ((2k + 1) θ) Olver, Frank W. NIST handbook of mathematical functions. Cambridge New York: Cambridge University Press NIST, 2010. p.226 f (t) = A cos (2πωc t) ( J0 (B) + 2 ∞ ∑ k=1 J2k (B) cos (2k (2πωm t))) −A sin (2πωc t) ( 2 ∞ ∑ k=0 J2k+1 (B) sin ((2k + 1) (2πωm t)))
cos a cos b = 1 2 (cos (a − b) + cos (a + b)) sin a sin b = 1 2 (cos (a + b) − cos (a − b)) ࡾ֯ؔͷੵެࣜ f (t) = AJ0 (B) cos (2πωc t) +2A ∞ ∑ k=1 J2k (B) cos (2k (2πωm t)) cos (2πωc t) −2A ∞ ∑ k=0 J2k+1 (B) sin ((2k + 1) (2πωm t)) sin (2πωc t)
ϕοηϧؔͷۙࣅ Tumakov, D.N. Lobachevskii J Math (2019) 40: 1725. https://doi.org/10.1134/S1995080219100287 The Faster Methods for Computing Bessel Functions of the First Kind of an Integer Order with Application to Graphic Processors Jn (x) = 2 (n − 1) x Jn−1 (x) − Jn−2 (x) ͜ͷۙࣅͷDPPMͳͱ͜Ζ ͭԼͷ࣍ ͱ ͭԼͷ࣍ ͕طͷ߹ ͷ͘͢͝؆୯ͳܭࢉͰ ͕ٻ·Δ Jn−1 (x) Jn−2 (x) Jn (x) Լͷ͔࣍Βܭࢉ͢ΔͱരͰϕοηϧؔͷ͕ٻ·Δ ∴
࣌ؒมԽ Τϯϕϩʔϓ؇͔ʹมԽ͢Δҝ ࣌ࠁ ʹ͍ͭͯٻΊͨ ࣌ࠁ ͷ ΛٻΊΔࡍͷॳظͱͯ͑͠Δ t B t + 1 B t t − 1 t − 2 t − 3 t + 1 t + 2 t + 3 Bt Bt Bt+1 Bt+2 Bt−1 Bt−2 ҰൠతͳFMԻݯ Λ࣌ؒͰมԽͤ͞ΒΕͳ͍ͷͰ ࣌ࠁ Ͱ࠷͍ Λग़ͨ͠ Λͦͷ··Ҿ͖ܧ͙ ωm t L ωm ωm ωm ωm ωm ωm ωm
Τϯϕϩʔϓਪఆ flinear (t, n, m) = (f (m) − f (n)) m − n t + nf (m) − mf (n) n − m fconst (t, n, m) = nf (m) − mf (n) n − m ferror (n, m, l) = ∫ n 0 flinear (t,0,n) − f (t) dt + ∫ m n flinear (t, n, m) − f (t) dt + ∫ l m fconst (t, m, l) − f (t) dt ͱ ʹ͍ͭͯͷޯΛٻΊ Λमਖ਼͍͚ͯ͠ྑ͍ n m ݮਰৼಈ͢Δָثͷ߹ Λ ʹ Λαϯϓϧݻఆ͢Δ fconst 0 l
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