Maths 1

Question 1

Cauchy-Riemann Equations

Answer 1

du/dx = dv/dy
du/dy = - dv/dx

Question 2

Ratio Test

Answer 2

lim[n→∞] |a(n+1)/(a(n))|

Question 3

Orthogonality Condition 1

Answer 3

∫ Cos(nx) * Cos(mx) dx = {π for m=n, 0 for m≠n

Question 4

Orthogonality Condition 2

Answer 4

∫ Sin(nx) * Sin(mx) dx = {π for m=n, 0 for m≠n

Question 5

Orthogonality Condition 3

Answer 5

∫ Cos(nx) * Sin(mx) dx = 0 (always)

Question 6

Dirichlet Conditions

Answer 6

• Single Valued
• Finite number of Discontinuities in one period
• Finite maxima and minima in one period
• Integral of the absolute value must be convergent

Question 7

Complex Fourier Series General Form

Answer 7

∑[n = -∞ to ∞] Cn e^(i2nπx/L)

Question 8

Complex Fourier Coefficient

Answer 8

Cn = 1/L * ∫[0-L] f(x) * e^(-i2nπx/L)) dx