mathPhys

for RUG, in 2023

40 flashcards · Physics · by Kristóf Nagy

1. sin -> e^

2. cos -> e^

3. geometric/power series sum

   if ∣r∣<1|r| < 1∣r∣<1

4. taylor series around a

5. Maclaurin series

6. binomial (1+x)k

7. binomial expansion/identity (x+y)n

8. cos(2x)

9. sin(2x)

10. cos(x)*cos(y)

11. sin(x)*sin(y)

12. δ(ak)=\delta(ak) = δ(ak)=

    δ(ak)=1∣a∣δ(k)\delta (ak) = \frac{1}{|a|} \delta(k)δ(ak)=∣a∣1​δ(k)

13. Fourier power series

    f(x)=∑n=−∞∞cneinπxLf(x)=\sum_{n=-\infty}^{\infty}c_n e^{i\frac{n\pi x}{L}}f(x)=∑n=−∞∞​cn​eiLnπx​

14. Fourier power series cn

    cn=12L∫−LLf(x)e−inπxLc_n =\frac{1}{2L} \int_{-L}^{L} f(x) e^{-i\frac{n\pi x}{L}}cn​=2L1​∫−LL​f(x)e−iLnπx​ dxdxdx

15. even function rule

    f(−x)=f(x)f(-x) = f(x)f(−x)=f(x)

16. odd function rule

    f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x)

17. even function examples

    cos(x),x2,x4,...cos(x),x^2,x^4,...cos(x),x2,x4,...

18. odd function examples

    sin(x),x,x3,...sin(x), x, x^3, ...sin(x),x,x3,...

19. odd integral rule

    ∫−aaodd=∫−aaodd∗even=0\int_{-a}^{a} odd = \int_{-a}^{a} odd * even = 0∫−aa​odd=∫−aa​odd∗even=0

20. Fourier series

    f(x)=a0+∑n=1∞ancos(nπxL)+∑n=1∞bnsin(nπxL)f(x)=a_0 + \sum_{n=1}^{\infty}a_n cos(\frac{n\pi x}{L}) + \sum_{n=1}^{\infty}b_n sin(\frac{n\pi x}{L})f(x)=a0​+∑n=1∞​an​cos(Lnπx​)+∑n=1∞​bn​sin(Lnπx​)

21. Fourier series a0

    a0=12L∫−LLf(x)dxa_0=\frac{1}{2L}\int_{-L}^{L} f(x) dxa0​=2L1​∫−LL​f(x)dx

22. Fourier series an

    an=1L∫−LLf(x)cos(nπxL)dxa_n=\frac{1}{L}\int_{-L}^{L} f(x) cos(\frac{n\pi x}{L})dxan​=L1​∫−LL​f(x)cos(Lnπx​)dx

23. Fourier series bn

    bn=1L∫−LLf(x)sin(nπxL)dxb_n=\frac{1}{L}\int_{-L}^{L} f(x) sin(\frac{n\pi x}{L})dxbn​=L1​∫−LL​f(x)sin(Lnπx​)dx

24. aux eq, 2 Re sol -> y_c

    yc=c1er1t+c2er2ty_c=c_1 e^{r_1 t} + c_2 e^{r_2 t}yc​=c1​er1​t+c2​er2​t

25. aux eq, 1 Re sol -> y_c

    yc=c1ert+c2terty_c = c_1 e^{rt} + c_2 t e^{rt}yc​=c1​ert+c2​tert

26. aux eq, 2 Im sol -> y_c

    yc=eat(c1cos(bt)+c2sin(bt))y_c = e^{at}(c_1 cos(bt) + c_2 sin(bt))yc​=eat(c1​cos(bt)+c2​sin(bt))

27. yp:eay_p: e^ayp​:ea

    yp=Aeay_p=Ae^ayp​=Aea

28. yp:x2exy_p: x^2e^xyp​:x2ex

    yp=(Ax3+Bx2+Cx)exy_p=(Ax^3+Bx^2+Cx)e^xyp​=(Ax3+Bx2+Cx)ex

29. ypy_pyp​ : cos x or sin x

    yp=Acos(x)+Bsin(x)y_p=Acos(x)+Bsin(x)yp​=Acos(x)+Bsin(x) (replace A and B with seperate x combination part if also x)

30. inverse Fourier transform

    f(x)=∫−∞+∞F(k)e2πikxdkf(x)=\int_{-\infty}^{+\infty} F(k)e^{2\pi ikx}dkf(x)=∫−∞+∞​F(k)e2πikxdk

31. forward Fourier transform

    F(k)=∫−∞+∞f(x)e−2πikxdxF(k)=\int_{-\infty}^{+\infty} f(x)e^{-2\pi ikx}dxF(k)=∫−∞+∞​f(x)e−2πikxdx

32. Dirac delta (k)

    δ(k)=∫−∞+∞e−i2πkxdx\delta(k)=\int_{-\infty}^{+\infty} e^{-i 2\pi kx}dxδ(k)=∫−∞+∞​e−i2πkxdx

33. Kronecker delta (nm)

    ∫−LLei(n−m)πLxdx=(2L)δnm\int_{-L}^{L} e^{i \frac{(n-m)\pi}{L}x} dx = (2L)\delta_{nm}∫−LL​eiL(n−m)π​xdx=(2L)δnm​

35. delta integral

36. delta(-x)=

37. |f(x)|2

    f(x)⋅f∗(x)f(x) \cdot f^*(x)f(x)⋅f∗(x) =∣f(x)∣2=|f(x)|^2=∣f(x)∣2

38. basic sin cos ^2

    sin2(x)+cos2(x)=1sin^2(x)+cos^2(x)=1sin2(x)+cos2(x)=1

39. euler's formula

40. geometric series sum