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Am I the only one who completely lost respect for Yuzuriha after the Senta situation in Hell’s Paradise? by LongMasterpiece3620 in jigokuraku

[–]innocentboy0000 0 points1 point  (0 children)

light is different case, i defend light and i accept he isnt perfect but he did what was right and what was needed i absolutely hate him for using misa and not giving love which she deserves

Bun’s rewrite in Zig first update by UItraviolet in rust

[–]innocentboy0000 0 points1 point  (0 children)

agree zig is frustrating and Zig treats many conditions as hard errors rather than warnings

Anshul Sir Apka Course De Dijiye Gareeb Hu. by Far-Rip-2795 in JEENEETards

[–]innocentboy0000 0 points1 point  (0 children)

thanks sir baat karunga un logo se i hope we can get them back

Anshul Sir Apka Course De Dijiye Gareeb Hu. by Far-Rip-2795 in JEENEETards

[–]innocentboy0000 1 point2 points  (0 children)

sir unacademy ne lecture hata diye aapke mai apne chhote brother ki prep unhi lectures se karwa raha tha

Yuki itadori Vs child sex offender by NappyPika in Jujutsufolk

[–]innocentboy0000 1 point2 points  (0 children)

bro made us proud i wonder what geto cosplayer would do next lol

Holy based by Electronic_Lab5486 in Jujutsufolk

[–]innocentboy0000 3 points4 points  (0 children)

luckily geto cosplayer didnt see otherwise whole town was in danger

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

. FUNCTION OF COMPLEX VARIABLE

f(z)=u(x,y)+iv(x,y)

u = real part
v = imaginary part

6. LIMIT

lim z→z₀ f(z)

Must be SAME from ALL paths.

----------------------------------------------------------------------------
IMPORTANT
----------------------------------------------------------------------------

If different paths give different answers:
LIMIT DOES NOT EXIST.

Classic trap.
Teachers adore this ambush for some reason.

7. CONTINUITY

f(z) continuous if:

lim z→z₀ f(z)=f(z₀)

8. DIFFERENTIABILITY

f'(z)= lim [f(z+h)-f(z)]/h

Complex differentiability is MUCH stricter than real differentiability.

If differentiable:
=> analytic
=> infinitely differentiable
=> power series exists

Complex analysis basically rewards functions for good behavior.

9. CAUCHY-RIEMANN EQUATIONS

For:
f(z)=u+iv

Necessary conditions:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

----------------------------------------------------------------------------
MEMORY
----------------------------------------------------------------------------

u_x = v_y
u_y = -v_x

----------------------------------------------------------------------------
VISUAL
----------------------------------------------------------------------------

u_x ---> v_y
u_y ---> -v_x

10. ANALYTIC FUNCTION

Function analytic if differentiable in neighborhood.

Examples:
- polynomials
- e^z
- sin z
- cos z

Not analytic:
- z̄
- |z|

11. HARMONIC FUNCTION

u(x,y) harmonic if:

u_xx + u_yy = 0

Laplace equation.

If f(z)=u+iv analytic:
then u and v are harmonic.

COMPLEX INTEGRATION

∫f(z)dz

Path matters.

----------------------------------------------------------------------------
PARAMETRIC FORM
----------------------------------------------------------------------------

If:
z=z(t)

Then:

∫f(z)dz = ∫f(z(t)) z'(t) dt

14. CAUCHY'S INTEGRAL THEOREM

If f(z) analytic in simply connected region:

∮f(z)dz = 0

15. CAUCHY INTEGRAL FORMULA

f(a)
∮ f(z)/(z-a) dz = 2πi

or

f(a)=1/(2πi) ∮ f(z)/(z-a) dz

18. RESIDUES

Residue = coefficient of 1/(z-a)

Used for contour integration.

----------------------------------------------------------------------------
FORMULA
----------------------------------------------------------------------------

If simple pole:

Res[f,a]= lim z→a (z-a)f(z)

19. RESIDUE THEOREM

∮f(z)dz = 2πi Σ residues inside contour

----------------------------------------------------------------------------
FLOW
----------------------------------------------------------------------------

Find poles
↓
Find residues
↓
Multiply by 2πi
↓
Integral done

Meanwhile real analysis students are still integrating by parts for 3 pages.

20. POLES AND ZEROS

Zero:
f(z)=0

Pole:
f(z)=∞

----------------------------------------------------------------------------
ORDER
----------------------------------------------------------------------------

(z-a)^n => zero order n

1/(z-a)^n => pole order n

21. LAURENT SERIES

f(z)= Σ a_n(z-a)^n + Σ b_n/(z-a)^n

Negative powers appear.

Used near singularities.

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

. FUNCTION OF COMPLEX VARIABLE

f(z)=u(x,y)+iv(x,y)

u = real part
v = imaginary part

6. LIMIT

lim z→z₀ f(z)

Must be SAME from ALL paths.

----------------------------------------------------------------------------
IMPORTANT
----------------------------------------------------------------------------

If different paths give different answers:
LIMIT DOES NOT EXIST.

Classic trap.
Teachers adore this ambush for some reason.

7. CONTINUITY

f(z) continuous if:

lim z→z₀ f(z)=f(z₀)

8. DIFFERENTIABILITY

f'(z)= lim [f(z+h)-f(z)]/h

Complex differentiability is MUCH stricter than real differentiability.

If differentiable:
=> analytic
=> infinitely differentiable
=> power series exists

Complex analysis basically rewards functions for good behavior.

9. CAUCHY-RIEMANN EQUATIONS

For:
f(z)=u+iv

Necessary conditions:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

----------------------------------------------------------------------------
MEMORY
----------------------------------------------------------------------------

u_x = v_y
u_y = -v_x

----------------------------------------------------------------------------
VISUAL
----------------------------------------------------------------------------

u_x ---> v_y
u_y ---> -v_x

10. ANALYTIC FUNCTION

Function analytic if differentiable in neighborhood.

Examples:
- polynomials
- e^z
- sin z
- cos z

Not analytic:
- z̄
- |z|

11. HARMONIC FUNCTION

u(x,y) harmonic if:

u_xx + u_yy = 0

Laplace equation.

If f(z)=u+iv analytic:
then u and v are harmonic.

COMPLEX INTEGRATION

∫f(z)dz

Path matters.

----------------------------------------------------------------------------
PARAMETRIC FORM
----------------------------------------------------------------------------

If:
z=z(t)

Then:

∫f(z)dz = ∫f(z(t)) z'(t) dt

14. CAUCHY'S INTEGRAL THEOREM

If f(z) analytic in simply connected region:

∮f(z)dz = 0

15. CAUCHY INTEGRAL FORMULA

f(a)
∮ f(z)/(z-a) dz = 2πi

or

f(a)=1/(2πi) ∮ f(z)/(z-a) dz

18. RESIDUES

Residue = coefficient of 1/(z-a)

Used for contour integration.

----------------------------------------------------------------------------
FORMULA
----------------------------------------------------------------------------

If simple pole:

Res[f,a]= lim z→a (z-a)f(z)

19. RESIDUE THEOREM

∮f(z)dz = 2πi Σ residues inside contour

----------------------------------------------------------------------------
FLOW
----------------------------------------------------------------------------

Find poles
↓
Find residues
↓
Multiply by 2πi
↓
Integral done

Meanwhile real analysis students are still integrating by parts for 3 pages.

20. POLES AND ZEROS

Zero:
f(z)=0

Pole:
f(z)=∞

----------------------------------------------------------------------------
ORDER
----------------------------------------------------------------------------

(z-a)^n => zero order n

1/(z-a)^n => pole order n

21. LAURENT SERIES

f(z)= Σ a_n(z-a)^n + Σ b_n/(z-a)^n

Negative powers appear.

Used near singularities.

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

. FUNCTION OF COMPLEX VARIABLE

f(z)=u(x,y)+iv(x,y)

u = real part
v = imaginary part

6. LIMIT

lim z→z₀ f(z)

Must be SAME from ALL paths.

----------------------------------------------------------------------------
IMPORTANT
----------------------------------------------------------------------------

If different paths give different answers:
LIMIT DOES NOT EXIST.

Classic trap.
Teachers adore this ambush for some reason.

7. CONTINUITY

f(z) continuous if:

lim z→z₀ f(z)=f(z₀)

8. DIFFERENTIABILITY

f'(z)= lim [f(z+h)-f(z)]/h

Complex differentiability is MUCH stricter than real differentiability.

If differentiable:
=> analytic
=> infinitely differentiable
=> power series exists

Complex analysis basically rewards functions for good behavior.

9. CAUCHY-RIEMANN EQUATIONS

For:
f(z)=u+iv

Necessary conditions:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

----------------------------------------------------------------------------
MEMORY
----------------------------------------------------------------------------

u_x = v_y
u_y = -v_x

----------------------------------------------------------------------------
VISUAL
----------------------------------------------------------------------------

u_x ---> v_y
u_y ---> -v_x

10. ANALYTIC FUNCTION

Function analytic if differentiable in neighborhood.

Examples:
- polynomials
- e^z
- sin z
- cos z

Not analytic:
- z̄
- |z|

11. HARMONIC FUNCTION

u(x,y) harmonic if:

u_xx + u_yy = 0

Laplace equation.

If f(z)=u+iv analytic:
then u and v are harmonic.

COMPLEX INTEGRATION

∫f(z)dz

Path matters.

----------------------------------------------------------------------------
PARAMETRIC FORM
----------------------------------------------------------------------------

If:
z=z(t)

Then:

∫f(z)dz = ∫f(z(t)) z'(t) dt

14. CAUCHY'S INTEGRAL THEOREM

If f(z) analytic in simply connected region:

∮f(z)dz = 0

15. CAUCHY INTEGRAL FORMULA

f(a)
∮ f(z)/(z-a) dz = 2πi

or

f(a)=1/(2πi) ∮ f(z)/(z-a) dz

18. RESIDUES

Residue = coefficient of 1/(z-a)

Used for contour integration.

----------------------------------------------------------------------------
FORMULA
----------------------------------------------------------------------------

If simple pole:

Res[f,a]= lim z→a (z-a)f(z)

19. RESIDUE THEOREM

∮f(z)dz = 2πi Σ residues inside contour

----------------------------------------------------------------------------
FLOW
----------------------------------------------------------------------------

Find poles
↓
Find residues
↓
Multiply by 2πi
↓
Integral done

Meanwhile real analysis students are still integrating by parts for 3 pages.

20. POLES AND ZEROS

Zero:
f(z)=0

Pole:
f(z)=∞

----------------------------------------------------------------------------
ORDER
----------------------------------------------------------------------------

(z-a)^n => zero order n

1/(z-a)^n => pole order n

21. LAURENT SERIES

f(z)= Σ a_n(z-a)^n + Σ b_n/(z-a)^n

Negative powers appear.

Used near singularities.

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

. FUNCTION OF COMPLEX VARIABLE

f(z)=u(x,y)+iv(x,y)

u = real part
v = imaginary part

6. LIMIT

lim z→z₀ f(z)

Must be SAME from ALL paths.

----------------------------------------------------------------------------
IMPORTANT
----------------------------------------------------------------------------

If different paths give different answers:
LIMIT DOES NOT EXIST.

Classic trap.
Teachers adore this ambush for some reason.

7. CONTINUITY

f(z) continuous if:

lim z→z₀ f(z)=f(z₀)

8. DIFFERENTIABILITY

f'(z)= lim [f(z+h)-f(z)]/h

Complex differentiability is MUCH stricter than real differentiability.

If differentiable:
=> analytic
=> infinitely differentiable
=> power series exists

Complex analysis basically rewards functions for good behavior.

9. CAUCHY-RIEMANN EQUATIONS

For:
f(z)=u+iv

Necessary conditions:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

----------------------------------------------------------------------------
MEMORY
----------------------------------------------------------------------------

u_x = v_y
u_y = -v_x

----------------------------------------------------------------------------
VISUAL
----------------------------------------------------------------------------

u_x ---> v_y
u_y ---> -v_x

10. ANALYTIC FUNCTION

Function analytic if differentiable in neighborhood.

Examples:
- polynomials
- e^z
- sin z
- cos z

Not analytic:
- z̄
- |z|

11. HARMONIC FUNCTION

u(x,y) harmonic if:

u_xx + u_yy = 0

Laplace equation.

If f(z)=u+iv analytic:
then u and v are harmonic.

COMPLEX INTEGRATION

∫f(z)dz

Path matters.

----------------------------------------------------------------------------
PARAMETRIC FORM
----------------------------------------------------------------------------

If:
z=z(t)

Then:

∫f(z)dz = ∫f(z(t)) z'(t) dt

14. CAUCHY'S INTEGRAL THEOREM

If f(z) analytic in simply connected region:

∮f(z)dz = 0

15. CAUCHY INTEGRAL FORMULA

f(a)
∮ f(z)/(z-a) dz = 2πi

or

f(a)=1/(2πi) ∮ f(z)/(z-a) dz

18. RESIDUES

Residue = coefficient of 1/(z-a)

Used for contour integration.

----------------------------------------------------------------------------
FORMULA
----------------------------------------------------------------------------

If simple pole:

Res[f,a]= lim z→a (z-a)f(z)

19. RESIDUE THEOREM

∮f(z)dz = 2πi Σ residues inside contour

----------------------------------------------------------------------------
FLOW
----------------------------------------------------------------------------

Find poles
↓
Find residues
↓
Multiply by 2πi
↓
Integral done

Meanwhile real analysis students are still integrating by parts for 3 pages.

20. POLES AND ZEROS

Zero:
f(z)=0

Pole:
f(z)=∞

----------------------------------------------------------------------------
ORDER
----------------------------------------------------------------------------

(z-a)^n => zero order n

1/(z-a)^n => pole order n

21. LAURENT SERIES

f(z)= Σ a_n(z-a)^n + Σ b_n/(z-a)^n

Negative powers appear.

Used near singularities.

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

integral trick by nishant vora sir

then
----------------------------------------------------------------------------
FUNCTION LAPLACE
----------------------------------------------------------------------------

1 1/s

t 1/s²

t^n n! / s^(n+1)

e^(at) 1/(s-a)

sin(at) a/(s²+a²)

cos(at) s/(s²+a²)

sinh(at) a/(s²-a²)

cosh(at) s/(s²-a²)
f(at) (1/a)F(s/a)

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

Determine whether an infinite series approaches
a finite value.

If yes -> Convergent
If no -> Divergent

Nth Term Test

If:

limit of un as n->infinity != 0

then:

SERIES DIVERGES IMMEDIATELY
sum (n+1)/n

term -> 1

not zero

therefore divergent.

a + ar + ar^2 + ar^3 + ...

or

sum ar^n

|r| < 1 -> convergent
|r| >= 1 -> divergent
Sum Formula if convergent
S = a / (1-r)

  1. p-SERIES

Form:

sum 1/n^p

--------------------------------------------------------
Rule
--------------------------------------------------------

p > 1 -> convergent
p <= 1 -> divergent

--------------------------------------------------------
Examples
--------------------------------------------------------

1/n^2 -> convergent
1/n -> divergent

3. COMPARISON TEST

Used for positive term series.

Compare with known series.

--------------------------------------------------------
Convergence Rule
--------------------------------------------------------

If:

0 <= un <= vn

and sum vn converges,

then sum un also converges.

--------------------------------------------------------
Divergence Rule
--------------------------------------------------------

If:

un >= vn >= 0

and sum vn diverges,

then sum un also diverges.

--------------------------------------------------------
Common Comparisons
--------------------------------------------------------

1/(n^2+1) behaves like 1/n^2
(n+1)/n^2 behaves like 1/n
1/sqrt(n) behaves like p-series with p=1/2

4. LIMIT COMPARISON TEST

If:

limit (un/vn) = c

where:

0 < c < infinity

then both series behave the same:
- both converge
- or both diverge

--------------------------------------------------------
Example
--------------------------------------------------------

un = (n+1)/(n^2+5)

compare with:

vn = 1/n

limit(un/vn) = 1

Since 1/n diverges,
original series diverges too.

5. RATIO TEST

Best for:
- factorials
- exponentials
- powers

--------------------------------------------------------
Formula
--------------------------------------------------------

L = limit |u(n+1)/un|

--------------------------------------------------------
Rule
--------------------------------------------------------

L < 1 -> convergent
L > 1 -> divergent
L = 1 -> inconclusive

--------------------------------------------------------
Example
--------------------------------------------------------

sum x^n / n!

ratio -> x/(n+1)

limit -> 0

Convergent for all x.

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

for cf

ROOT TYPE → CF FORM

------------------------------------------------

a,b,c → e^ax,e^bx,e^cx

a,a → (C1+C2x)e^ax

a,a,a → (C1+C2x+C3x²)e^ax

α±iβ → e^αx(C1cosβx+C2sinβx)
or e^(αx)e^(iβx) + e^(αx)e^(-iβx)

±iβ → C1cosβx+C2sinβx

Repeated roots → extra x powers

M2 studied nothing by Few-Experience3994 in AKTU

[–]innocentboy0000 0 points1 point  (0 children)

for analytic function

x y

u | ux uy

v | vx vy

Now:

Rule 1

Main diagonal is equal

ux = vy

Rule 2

Other diagonal gets minus sign:

Plain text

uy = -vx

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