USEFULMATHEMATICALRELATIONS

USEFULMATHEMATICALRELATIONS - USEFUL MATHEMATICAL RELATIONS...

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Unformatted text preview: USEFUL MATHEMATICAL RELATIONS A) Coordinate Transformations: Rectangular–to–Spherical (and Vice–Versa): ax = aR sin (θ ) cos (ϕ ) + aθ cos (θ ) cos (ϕ ) − aϕ sin (ϕ ) a y = aR sin (θ ) sin (ϕ ) + aθ cos (θ ) sin (ϕ ) + aϕ cos (ϕ ) az = aR cos (θ ) − aθ sin (θ ) aR = ax sin (θ ) cos (ϕ ) + a y sin (θ ) sin (ϕ ) + az cos (θ ) aθ = ax cos (θ ) cos (ϕ ) + a y cos (θ ) sin (ϕ ) − az sin (θ ) aϕ = − ax sin (ϕ ) + a y cos (ϕ ) Rectangular–to–Cylindrical (and Vice–Versa): ax = ar cos (ϕ ) − aϕ sin (ϕ ) a y = ar sin (ϕ ) + aϕ cos (ϕ ) az = az ar = ax cos (ϕ ) + a y sin (ϕ ) aϕ = − ax sin (ϕ ) + a y cos (ϕ ) az = az Cylindrical–to–Spherical (and Vice–Versa): aR = ar sin (θ ) + az cos (θ ) aθ = ar cos (θ ) − az sin (θ ) aϕ = aϕ ar = aR sin (θ ) + aθ cos (θ ) aϕ = aϕ az = aR cos (θ ) − aθ sin (θ ) B) Trigonometric Relations: sin (α ∓ β ) = sin (α ) cos ( β ) ∓ cos (α ) sin ( β ) cos (α ± β ) = cos (α ) cos ( β ) ∓ sin (α ) sin ( β ) ⎛π ⎞ ⎛π ⎞ sin ⎜ ∓ α ⎟ = cos (α ) , cos ⎜ ∓ α ⎟ = ± sin (α ) ⎝2 ⎠ ⎝2 ⎠ sin (α ) cos ( β ) = 1 ⎡sin (α + β ) + sin (α − β ) ⎤ ⎦ 2⎣ cos (α ) sin ( β ) = 1 ⎡sin (α + β ) − sin (α − β ) ⎤ ⎦ 2⎣ cos (α ) cos ( β ) = 1 ⎡cos (α + β ) + cos (α − β ) ⎤ ⎦ 2⎣ 1 sin (α ) sin ( β ) = − ⎡cos (α + β ) − cos (α − β ) ⎤ ⎦ 2⎣ sin ( 2α ) = 2 sin (α ) cos (α ) cos ( 2α ) = 2 cos 2 (α ) − 1 = cos 2 (α ) − sin 2 (α ) = 1 − 2sin 2 (α ) tan ( 2α ) = 2 tan (α ) 1 − tan 2 (α ) e ± jα = cos (α ) ± j sin (α ) , where j is the unit imaginary number. e j α − e− jα sin (α ) = 2j e j α + e− j α 2 cos (α ) = C) Useful Integrals: e ax ⎡ a sin (bx + c ) ∫ a2 + b 2 ⎣ Since C = 0.577215665 is the Euler constant, e ax sin (b x + c ) d x = x sinu du = x u Si ( x ) = ∫ 0 ∞ Ci ( x ) = cosu du = C + ln ( x ) u ∫ x x Cin ( x ) = ∫ 1 0 Ein ( y ) = jy ∫ 0 x3 x5 + 3 × 3! 5 × 5! 1 cosu du = C + ln ( x ) u e w w b cos (bx + c ) ⎤ ⎦ .... x2 x4 + 2 × 2! 4 × 4! Sine Integral .... Cosine Integral Ci ( x ) dw = Cin ( y ) + j Si ( y ) Exponential Integral E i {± j u } = Ci (u ) ± j S i (u ) Also; ( −1) x 2 k +1 Si ( x ) = ∑ k = 0 ( 2k + 1)( 2k + 1) ! k ∞ ∞ Ci ( x ) = C + ln ( x ) + ∑ ( −1) k =1 ∞ Cin ( x ) = ∑ ( −1) k +1 k =1 k x2k ( 2k )( 2k )! x2k ( 2k )( 2k )! D) Hyperbolic: e α − e −α α3 α5 α7 =α + + + + ... 2 3! 5! 7! e α + e −α α2 α4 α6 = 1+ + + + ... cosh (α ) = 2 2! 4! 6! cosh 2 (α ) − sinh 2 (α ) =1 sinh (α ) = cosh ( x ± j y ) = cosh ( x ) cos ( y ) ± j sinh ( x ) sin ( y ) sinh ( x ± j y ) = sinh ( x ) cos ( y ) ± j cosh ( x ) sin ( y ) E) Logarithmic: log b ( M N ) = log b ( M ) + log b ( N ) log b ( M / N ) = log b ( M ) − log b ( N ) log b ( M n ) = n log b ( M ) log a ( N ) = log b ( N ) log a ( b ) = log b ( N ) / log b ( a ) log e ( N ) = ln ( N ) = log10 ( N ) log e (10 ) = ln (10 ) log10 ( N ) = 2.302585 log10 ( N ) log10 ( N ) = log e ( N ) log10 ( e ) = 0.434294 log e ( N ) = 0.434294 ln ( N ) F) Taylor Series: df f ( x ) = f ( x0 ) + ( x − x0 ) dx ex = 1 + x + 2 3 x x + + ... 2! 3! + x = x0 ( x − x0 ) 2! 2 d2 f dx 2 + ... x = x0 1 = 1 + x + x 2 + x3 + ... for x < 1 1− x x x2 1 + x = 1 + − + ... for x < 1 28 sin ( x ) = x − x3 x5 x 7 + − + ... 3! 5! 7! x2 x 4 x6 cos ( x ) = 1 − + − + ... 2! 4! 6! F) Others: (1 + x ) n = 1+ n x + n ( n − 1) 2 n ( n − 1) ( n − 2 ) 3 x+ x + ... 2! 3! for x < 1 Binomial Expansion ∞ ⎛ 2π n ⎞ ∞ ⎛ 2π n ⎞ f ( x ) = a0 + ∑ an cos ⎜ x ⎟ + ∑ bn sin ⎜ x⎟, ⎝ T ⎠ n =1 ⎝T ⎠ n =1 1 a0 = T 2 an = T 2 bn = T T2 ∫ f ( x ) dx −T 2 T2 ∫ −T 2 ⎛ 2π n ⎞ f ( x ) cos ⎜ x ⎟ dx ⎝T ⎠ Fourier Coefficients T2 ⎛ 2π n ⎞ ∫ f ( x ) sin ⎜ T x ⎟ dx ⎝ ⎠ −T 2 References: 1. Balanis, C. A., Antenna Theory, Analysis and Design, John Wiley & Sons Ltd., New York, 1997. 2. Canbay, C., Anten ve Propagasyon I, Yeditepe University Press, İstanbul, 1997. 3. Pozar, D. M., Microwave Engineering, John Wiley & Sons Inc., New York, 1998. 4. Kreyszig, E., Advanced Engineering Mathematics, Wiley, 9th Ed., 2005. ...
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This note was uploaded on 10/16/2009 for the course EL el6303 taught by Professor Prof during the Spring '09 term at NYU Poly.

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